why do you think that these yeshivas will stay immune to this?
well there's the "G-d Factor".
Hashem doesn't like YU too much.
Before anyone kills me. I'm not saying the people who teach there are bad and im not saying that the students learning there are bad. But...
YU, the school, has been going downhill for a long, long, time. It went from a place where many of the Gedolim of last generation learned to a placed steeped in apikorsus, progressivism, and a whole lot of garbage. I can say with a certainty that YU does not have many brownie points with Hashem and it doesn't have that much longer before it goes completely off and becomes Chazzer Treif or shuts down.