In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm cf}(\aleph_1)=\aleph_0$, and most conceivable limit alephs also have cofinality $\aleph_0$. I recall reading in a book that it is "unknown" if there are any ordinals that are provably of uncountable cofinality in ZF, so this is really a reference request for progress on this problem. Is this a "provably unprovable" problem? Are there any large cardinal hypotheses (other than AC) that shed some light on the problem?

No. As Miha quotes, it is consistent (relative to some very very large cardinals) that no initial ordinal (read: $\aleph$ number) has an uncountable cofinality. Since the cofinality of an ordinal is always an initial ordinal, this finishes the proof.

Note that very large cardinals are necessary. If $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$ then there is an inner model with a Woodin cardinal. So to have all the cardinals with countable cofinality you have to expect some proper class of very large cardinals.

Gitik proved this from a proper class of strongly compact cardinals, which is quite a large assumption. (Note, however that "proper class of ..." is quite scary, but still weaker from something like "inaccessible cardinal which is a limit of ...")