A "true" sandbox, IMO, will have a LOT of improv by the DM because the players will often go off to places, etc. which the DM has not prepped beforehand.
Yes, the DM can present multiple hooks, each one prepared ahead of time to one degree or another, but if they players decide not to follow any of them, and are free to go do something else (which the DM has not prepared at all), then it becomes improv on the DM's part.
That isn't to say that once the PCs have decided on a path, the DM can prepare for that next session, but in the meanwhile it is improv.
Another is the "fake" sandbox. The DM has a planned adventure (say a dungeon) but the players do something else. Later on, the DM "recycles" the un-used dungeon/adventure elsewhere. The players never (or not often) realize that it was already made and just needed a bit of tweaking to get up to speed later on. Frankly, I do this often. After all, I put a lot of time and effort into the adventure, and to "not play it at all" seems a colossal waste of time and I find it frustrating.
To be clear, this is not the same as presenting two or three hooks and the players returning to an unexplored hook later on.
I ask that the players try to decide where they're headed before the session so I can prep. Even then they often make unexpected choices and I have to adjust. Because of that I'll usually have more encounters sketched out than I need and change the descriptions of the enemy to fit the new scenario. I also typically don't worry about detailed maps because it's just not worth it. Most video games by their very nature are pretty linear, if you skip a quest in Meridian it will frequently be there later on. But that's not what happens in the games I run. Some hooks may be available later on, some may be similar but changed based on the actions of the characters, many hooks will simply fade away.
I have no issue with more linear games, they can be just as fun as a player as long as I know what I'm getting into as long as there's a bit of wiggle room.