What's Wrong with Hume's Fork

In Section 4 of his Enquiry Concerning Human Understanding, David Hume makes a famous distinction between "matters of fact" and "relations of ideas"

All the objects of human reason or enquiry fall naturally into two kinds, namely relations of ideas and matters of fact. The first kind include geometry, algebra, and arithmetic, and indeed every statement that is either intuitively or demonstratively certain. That the square of the hypotenuse is equal to the squares of the other two sides expresses a relation between those figures. That three times five equals half of thirty expresses a relation between those numbers. Propositions of this kind can be discovered purely by thinking, with no need to attend to anything that actually exists anywhere in the universe. . . . Matters of fact . . . are not established in the same way; and we cannot have such strong grounds for thinking them true. The contrary of every matter of fact is still possible, because it doesn't imply a contradiction and is conceived by the mind as easily and clearly as if it conformed perfectly to reality. That the sun will not rise tomorrow is just as intelligible as - and no more contradictory than - the proposition that the sun will rise tomorrow.

Note Hume's claim that "All the objects of human reason or enquiry fall naturally into two kinds, namely relations of ideas and matters of fact." This claim is known today as Hume's Fork. Relations of ideas are subject to strict demonstration or proof, but imply nothing at all about what exists. Matters of fact, on the other hand, are not subject to strict demonstration or proof but do tell us something about what exists. So far so good.

But what about Hume's Fork itself? It too is an object of "human reason or enquiry". Accordingly, it too must either be a relation of ideas or a matter of fact. Which is it?

It does not seem to be a relation of ideas, since it's denial does not obviously entail a contradiction. Nor does it seem to be a matter of fact, since Hume presents the Fork as though it were an a priori truth knowable independently of experience. Hume's Fork is, therefore, a prima facie counterexample to itself. Given the importance of the Fork in Hume's thought, this suggests that there's something amiss at the very foundations of Hume's philosophy.