Ok, let's stop this RIGHT HERE. An argument over whether or not religious leaders are nuts is a quick way to get a thread permanently locked. Thank you.
Now, my thoughts on this:
The Emperor almost certainly has this:
https://en.m.wikipedia.org/wiki/Hypertrichosis
It's a very real medical condition, but historically speaking people with it were thought to be some sort of evil. Sophia probably intends to reveal him in front of the entire synod. The consequences of the Roman Emperor being compromised in such a way will be disastrous. I don't think either the Konmenos or the Dalassenos have gauged her correctly; unless the Konmenoi want the Emperorship back... Which they might.