Restating your prior point in a different way doesn't make it any more or less correct. The point is these two things seem to be independent from each other, which, if true, would already disprove the modified claim you are presenting.

The issue is, there exist plenty of people who are bad at both, good at both, and bad at one and good at the other. This pattern doesn't support a strong connection between being class conscious and being socially conscious.