3 Eduard Čech was not Polish

It is well known that the Stone-Czech 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?

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# Is choice needed to establish the existence of idempotent ultrafilters?

It is well known that the Stone-Czech 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?