A unique $Q$-point and infinitely many near-coherence classes of ultrafilters

We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-${\omega }_2$ countable support iteration of restricted Mathias forcing over a ground model satisfying $\mathsf{\text{CH}}$, there is a unique $Q$-point up to isomorphism. In particular, it is consistent that there is only one $Q$-point while there are $2^{\mathfrak{c}}$-many near-coherence classes of ultrafilters.