At least Conservatives don't reject the idea that objective truth exists.
But, I agree that what Walker is doing is bad, although the Left is mostly responsible. Very few universities and academics believe in objective truth, and yet they still want funding for humanities departments, chairs, and professorships. For what purpose do humanities departments exist if truth is not held to be real and in some sense searchable and discernable? The answer for the past 60 years has been to espouse radical Leftism.
There are (have been) only two reasons that provide a compelling incentive and impetus to fund the study of the humanities on a societal level. First is if society generally agrees that truth exists and its pursuit is therefore necessary. This is what a Medieval person would have felt, and is precisely why the university in the West was established in the first place. Second is the notion that humane learning is necessary for freedom, for how is man free if he cannot think about what the good life is and how and whether to pursue it. This is how the ancient Greek would have felt. A combination of these two gave rise to the study of the liberal arts in America, and is why until the past few decades Americans had a unique devotion to humane learning.
But, now, there are too many Americans like the OP and not enough Americans like me to justify pouring money into the study of humane things, which is why the humanities are largely dying. Walker and Perry would rather just ditch the humanities from the university altogether, which is sad, but better than the status quo (if we aren't going to be truly free and liberally educated, at least let's not pretend in our hubris that we are). A truly authentic Conservative solution would be to return to a truly liberal arts curriculum, which means studying Western literature, philosophy, theology, theatre, history, and government from Homer until the present. Put differently, the solution would be to fire all the professors who teach classes like "Britney Spears and 1990s music", "queer ecology", "feminist popstars," and "Harry Potter", and hire professors who would teach courses like "Ancient Greek Tragedy", "Ancient Greek Comedy", "Enlightenment Philosophy", "Dostoevsky and Nietzsche," etc...
But, alas, that would demand that we actually care about searching for truth, something that the majority of college-educated liberals do not believe in in the first place. As the New York Times described yesterday, the "Common Core" is trying to teach 2nd graders that all opinions cannot be facts (only scientific material claims can be factual). It's basically just conservatives and poor minorities who still cling to things like guns, religion, and "truth". In such a society, the university has lost its raison d'etre.