> How do I tell my colleagues to stop contributing unverified AI output without creating tension between us?

Embrace the tension. Tension is human.

Out of work I wouldn't mind, but I spend 8h/day there and I am forced to work with these people, so I'd prefer to keep the drama out so that I can focus on solving problems.

The other person already demonstrated a lack of professionalism by sharing unverified AI slop so, in case of conflict, I wouldn't be surprised if they continued acting unprofessionally by spreading false rumors, unnecessarily escalating the situation to higher ups, secretly sabotaging the project, etc.