They may have meant conservative in an older sense of wanting to change things, but slowly. American “conservatives” are more properly called reactionaries because they neither want to keep things the way they are, nor incremental progress (both of which fall under conservativism), but rather want to actively roll back and take away rights that have already been won. Liberals are conservatives because they want gradual, incremental change while preserving the social order. Conservativism is still right-wing though so that part of their statement is incorrect.

Did they not confirm that because they’d immediately show themselves to be wrong?

Do democrats not view a hierarchical social order as inevitable, natural, normal, or desirable? Capitalism is a hierarchical social orders and liberals definitionally support capitalism.