I mean the player is being asked to do it. The DM isn't doing it for the player. The player can say whatever they want as "true" or "false," and if they only want true rumors, just tell the DM?

I'm probably missing something, but I'm confused as to how asking a player for some additional information in their character building that gets shared with the group to facilitate the game can be triggering in a way that goes beyond the game. I really do hate the "if you don't like it, don't play" response, but there are some things where the trigger is inherent to the activity.

Gamers™ are known for not having a nuanced understanding of mental health (because, y'know, the only emotion men have is anger) and we're just seeing it here. Thankfully I have a hairtrigger block button :)