So I managed to get Ulfric and Tullius to agree to the peace conference at High Hrothgar. I got there, spoke with the Graybeard leader, who told me to take my seat in the council hall and the conference would begin. But I went into the hall and took my seat, and nothing happened! Everyone just stood there, looking at each other, but when I got up to talk to them no one would say anything noteworthy to me. I tried getting up and sitting back down a few times, and also reloading a game from before I initiated the "Take your seat" sub-objective with the Graybeard leader. No luck. Obviously, this breaks the main quest if I can't proceed from here.

Has anyone else encountered this, and is there a way around it?