Cardiff & Klein’s study, as others have pointed out, has two major problems: (1) They looked only at a very small number of colleges in California—including oversampling UC schools at the expense of CSU schools, and ignoring community colleges altogether—and (2) that they analyzed party affiliation, which isn’t actually a good stand-in for sociopolitical leanings, and also excluded individuals with no party affiliation from their ratios despite the fact that (as they mention, but only in passing) those made up a majority of faculty members.
Really, there are no good studies on this, at least as far as I’ve been able to find.