It is well known that the Stone–Čech compactification $\beta \mathbb N^+$ of the positive natural numbers has the structure of a compact left semitopological semigroup and hence, by Ellis's lemma, has idempotents. The usual proof of Ellis's lemma uses Zorn's lemma. Idempotent ultrafilters are clearly non-principal. It is known that the existence of non-principal ultrafilters is weaker than the axiom of choice.

My question is whether the existence of idempotent ultrafilters in $\beta \mathbb N^+$ is still weaker than choice?