This still does not provide any proof for the claim. Of course Marxists want to promote their ideology, even culturally. Nobody denies this. The claim made by people using the term "Cultural Marxism", and the reason it is a conspiracy theory, is that said Marxists do so for the purpose of destroying Western culture.
Again, to be clear: I am not asking for proof that Marxists promoted Marxism (obviously they did), but that they did so "to subvert Western culture".