Crowded rational ultrafilters

We prove that if every family in (ωω,≤∗) of size less than c is bounded then there exists a point p in Q∗ such that p generates an ultrafilter in the set-theoretic sense on Q and such that p has a base consisting of sets that are homeomorphic to Q. This is a partial answer to Question 30 (Problem 229) in (Hart and van Mill, 1990).