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Yes, it's still weaker. To build a model of ZF in which choice fails but $\beta\mathbb N^+$ has idempotents, start with a model of ZFC (which will, of course, have idempotent ultrafilters in $\beta\mathbb N^+$). Add a lot of Cohen-generic subsets of some regular cardinal $\kappa$ well above the cardinal of the continuum; forcing conditions are partial functions of size $<\kappa$. No new reals are added, and your idempotent ultrafilters from the ground model are still idempotent ultrafilters (and AC still holds). Now pass to the symmetric submodel given by the group of automorphisms of your forcing that permutes the names of the added Cohen subsets, with the filter determined by supports of size $<\kappa$. That model violates choice, because you can't well-order the power set of $\kappa$. But the ground model's reals and ultrafilters haven't been touched, so you still have the same idempotent ultrafilters that you had to start with.