Not with what a PC thinks or does. The rules give that power to the player.
So far as I can make out, that misreads the wording in question. Those words define what it is to be roleplaying. They do not impose any constraint on the DM.
My belief that it is certain is not an assumption, it’s the conclusion I’ve drawn from my reading of the rules, such as the statement under discussion that a player decides what their character does. To establish uncertainty, the DM must suspend that decision-making power, which I do not see support in the rules for them doing.
Ah, ok, I see the confusion here. I too am arguing the premise as presented
by @clearstream as a means of resolving the conflict with the “roleplaying rule.” They presented the argument that the “roleplaying rule” represents a dichotomy between roleplaying - wherein the player decides what their character thinks, says, and does - and mechanics, wherein the rules say what the PCs think, say, and do. They suggested that the DM has the authority to suspend roleplaying (as they are defining it here) in order to rule that an action made with the intent of forcing a PC to make a particular decision has an uncertain outcome. Rather than dispute the premise of this position, I argued it as presented, saying that I do not see a suggestion in the rules that the DM ought to “suspend roleplaying” in order to get around the “roleplaying rule.” Is that a bit clearer?
I don't suggest a dichotomy: it is not all or nothing. But look, you continue asking for a special reference to a DM's ability to decide a game mechanic overrides player decisions over what their character thinks, says, and does (suspending roleplaying in that respect.) In doing so, are you able to explain why you ignore how the game is ordinarily played? The DM is
incessantly overriding player decisions over what their character thinks, says, and does: if a special carve out were required, it would be for social skills. And I have an idea about why might not notice that.
Thinking about
@Swarmkeeper's normal room, here is an attempt to illustrate the idea. A player C creates a fighter and reaches 3rd level, choosing Champion (because as you will see, they are a champ!) The following exchange ensues:
C: I fly over to where the dragons are and...
DM: Um, look, your character cannot fly, maybe if...
C: But don't I get to decide what my character does (points at PHB175)
DM: ...
C: Okay, well if I can't fly there I teleport to...
DM: Sorry, but you also do not have the ability to teleport, would you...
C: Right, I dematerialise the dragons from here then, by thinking at them
psionically!
DM: ...
Later
DM: The dwarf claims to know nothing about it, but...
C: I want to know if he's telling the truth!
DM: Okay, you can use your Insight against the dwarf's Deception
C: That's not right - I decide what my character thinks!
DM: Yes, but - your Insight is pretty good and he's willing to answer your, perhaps if you just roll...
C: Nope, there's no uncertainty here, he's lying!!
DM: ...
Much later, C finally encounters a dragon
DM: Okay, so you failed your Wisdom saving throw against Frightful Presence and..
C: Oh no, nope, no way. I decide what my character thinks, says and does, and my character is not frightened - it doesn't affect me. Chaaarrrrge!!!
DM: ...
What I believe you are overlooking is that DMs constantly, as a matter of necessity, in order that the game can function
at all override what players decide their characters think, say and do. So far as I can make out, that blindness is caused by compartmentalizing every other case where that happens, from a small set of cases under social interaction. Or here is the question: how do you believe the game works - how are game mechanics applied - if a DM is not expected to ordinarily override a player's freedom to decide their character thinks, says, or does (judiciously suspending some facet of roleplay, in each case)?
The note of difference - and this is highlighted by the normal room example - is that outside of social interaction players habitually go along with that overriding, so that in play it feels more like a constraint on what they
choose to do rather that a suspension of roleplay even though it is
founded upon a suspension of roleplay. Players don't choose to say that dragon fear doesn't affect them (when they fail their saves), even though it is deciding a fact about how their character thinks.
And that is the difference that I find you and some others here somehow oblivious to. My silly example has a serious intent: I hoped to show how dysfunctional the game will be if a player doesn't come into it with that continuous concession to mechanics already in mind. And that is what is happening in regard to expectations around social interaction: hence I say that they're smuggled into a special category.
I just used that site. In the entire MM there are 8 creatures with intimidate, and half of those are orcs and ultroloths, and 8 with persuasion, 2 of which are Mind Flayers, and 4 of which are dragons. Only 2 other creatures in the entire book have it. And there are over 300 monsters in the book.
In case doubts arise about this, I count 76 MM monsters with one or more of deception, intimidation or persuasion. Using DNDBeyond. Three pages of 20 monsters each, and one of 16. Example, Bone Devil, deception +7, or Adult Green Dragon, deception +8, persuasion +8.