Of course it will. And what happens then? Degrees from 'Elite' institutions will be worth more, and the price will continue to go up as those that can afford it, fight for places.
And how's that any difference to K-12 education???
I know well off families (a couple of RE agents) that send their kids through the public system as they see no value in paying more and others who are fairly middle income families who work 1.5 jobs each, got their kid a scholarship to my daughter's top private school and scrimp and save every cent.
The same will happen if this occurs for universities. People will still make choices.