But you haven't answered the underlying question. Does Francis the Guard exist? Can the player track them down in that town, now that they have pulled that from their backstory?
You've already called me on this, so yes I know the question isn't for me, but, for my part at least, I'm happy to say that Francis the Guard exists (or at least did exist). I'm even happy to go with any reasonable suggestion regarding the existence of any NPC implied to exist by the backstory. What I'm not happy about is a player dictating to me that a particular NPC is Francis, or something like that Francis is a 12th level fighter, or what have you. I might decide that those things make for an interesting game or at least is reasonable for the games demographics*, but I can do that as an impartial judge. A player, because they are managing an avatar in the game world and trying to "win", doesn't have the impartiality to decide that.
*(This guard is much more likely to be Francis, if this is a village of 80 people, and only has one guard, than it is to be Francis in a city of 50,000 people which has a town watch of 500 individuals, etc.)
So, at my table, I don't necessarily disagree with any of this. I would say that it is a bit harsh to lock a player into only the backstory they come up with before play begins, but only because I often have players who can't come up with a backstory until two or three sessions in. In fact, even on fairly robust backstories, I, myself, and my players have found new inspiration which led to refining and adding details to those. So, knowing my full backstory is locked after session one is fair, but not something I would do personally.
I find that players that don't make backstories aren't interested in backstories and generally don't have literary goals for their character - that is to say, they don't really care if the character is involved in any sort of narrative arc except supporting the "plot" and meta-plot of the campaign. The characters growth as a person is not interesting to them compared to the characters growth as a playing piece. And that's OK.
I strongly encourage long backstories even if I don't always get more than a few sentences. Adding to a backstory at any point generally requires mutual consent of both the GM and the player, although typically when a character is created I ask the player how much they are willing to have me "mess with them", by which I mean introduce complications based on their backstory and even elements of their backstory which they themselves didn't realize ("you are an illegitimate child", "you are adopted", "your family is suffering from a curse", "turns out, you aren't even human", etc.). Some players want me to run wild and introduce complications. Others feel that if you do that, you are bullying them. I try to accommodate both desires.
However, if you cannot tell a player what they think, which was iserith's position both in this thread with the orc elder telling stories about monster weaknesses and the insight thread, then even getting to this point can be troublesome. Because the player may have established that the guard named Francis does exist.
IF we cannot ever tell a player what they think, and they state "I once saved this Red Dragon's life by healing it of Dragon Pox" then we have a disconnect in the game reality. The player believes this, something must have triggered this belief, but the DM says it never happened. So why does the player have these memories? This is where the "false dilemma" you see comes from.
You probably aren't doing this on purpose, and I've certainly used the terms inappropriately a ton of times, but for this passage the difference between the player and the character really matters and I can't be absolutely sure which you mean. Does the character believe that the guard is Francis, or does the player believe that the guard is Francis?
If the player believes that the guard is Francis, we have an out of game problem that has to be addressed as an out of game problem. Somehow the player got confused as to the fictional state of the game world or his role as a player, and we have to iron that out - just as if the player started performing a plan, and I have to explain something like, "You did understand that there is a deep chasm running across the middle of the room." and it turns out that they thought it bisected the room in a different manner, or if the player starts outlining a plan and I have to explain, "I'm not sure if I made this clear, but the thing I described is about 400 yards away."
But if the character believes that the guard is Francis, then I am not telling the player how to play their character or what the character believes. They are free to tell me that the character believes that the guard is Francis and has whatever memories that he wants to have regarding "Francis". And they are free to invent their own reasons why the character thinks as they do or however they want to rationalize this false belief.
Again, describing the setting as it actually is - no matter how you twist words around - is not telling the player what to think or how to play their character. It's just trying to communicate as many relevant facts as the player needs to make a judgment of how they want to play. The player can no more tell me that the NPC is Francis the Guard than they can tell me that the chest actually contains a fortune in gems. If the player asserts, "My character believes that this is Francis the guard" or "My character believes that the chest contains a fortune in gems.", that's swell, but the characters belief doesn't make the asserting true. A player can ask, "Hey, according to my backstory I grew up in this town, and was friends with a young man named Francis that became a guard. Is Francis among the guards?" And that's a perfectly valid question, and the answer may be, "Yes. Yes he is." Luke's player can say, "Heh, my friend Biggs Darklighter wanted to join the Rebellion, and this is a rebel base. Is he here?", and that might make a great call out. But you can't assert things like, "Biggs Darklighter is here", "Biggs Darklighter is the base commander.", or whatever you want and expect them to be true just because you say that the character thinks it is true.
And if you do, it's not my job to explain why the character thinks it is true, or to get the player to back down. It's not my character.
Honestly, part of what drew me into this example was how close it was to the Elder telling the character how to slay various monsters when they were a child, which everyone on one side accepted this was perfectly fine, but this example raised an outcry of players far overstepping their bounds and declarations they would be better off playing a different game. The difference between the two, in a narrative sense, is minimal. The only difference is one establishes knowledge a player likely already had and would use in fights, and the other gives them a social benefit in a situation.
Heck, I'm not even aware of that argument or all the agendas that lie behind this thread. For my part, you can assert that someone told your character how to slay various monsters when you were a child, and I consider that a perfectly fine thing for you to assert. Trouble is, it doesn't change the fact that you will get no special treatment from me unless your character also has spent CharGen resources on whatever lore skills are necessary to actually learn facts about monsters. If you, as a player of a character that has no lore skill regarding monsters, assert that your character was told a lot of stuff, the very fact that your character does not have a bunch of points spent on monster lore proves that what you were told was probably incomplete, or common knowledge, or just plain wrong. The facts on your character sheet disprove your claim of special knowledge - or else they don't. That call out to your background may perfectly explain why you do have all that lore on your character sheet. Or it may just explain that natural 20 you rolled to identify this particular monster.
If your character has BAB +0, telling me how you were taught for years by a swordmaster how to fight in your youth, won't convince me to give you a +20 bonus on attacks. It just convinces me you weren't a very good student because the facts say you aren't a great sword master.
Or else it convinces me you are a problem player that is going to require special handling to deal with your emotional needs.