Will a grade school and high school steal students from my university?

This question is related to this other question: Once you build a University, can the high school and community college be safely closed?

I find that my University is routinely under-used, while I have two grade schools that are routinely close-to-full.

Has anyone experimented with disabling their grade schools? Does the university fill up? Or are the student pools entirely different?

Update: I tried powering off my grade schools and high school during the night (community college was already off). Around 4pm of the that day, the University crossed the 1,000 students threshold.

So I think this is evidence for the "single student" pool hypothesis (which is weird I suppose).