TITLE: What are some good lower bounds on the consistency of the failure of the PCF conjecture?
QUESTION [13 upvotes]: Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$.
But the conjecture is that $\omega_4$ can be provably replaced by $\omega_1$. Namely, $2^{\aleph_\omega}<\aleph_{\omega_1}$ holds, assuming that $\aleph_\omega$ is a strong limit cardinal.
As far as I understand it, we know that from large cardinal assumptions it is consistent that $2^{\aleph_\omega}$ is arbitrarily large below $\aleph_{\omega_1}$ (and it is a strong limit, of course). But there is no current way to go beyond $\aleph_{\omega_1}$. Even Gitik's work on the subject does not translate to the $\aleph_n$'s.

Question. Suppose that the PCF Conjecture fails. Namely, $\aleph_\omega$ is a strong limit cardinal, but $2^{\aleph_\omega}>\aleph_{\omega_1}$. What kind of large cardinals can we expect to find in inner models?

(Of course large cardinals are necessary, since $2^{\aleph_\omega}>\aleph_{\omega+1}$ with $\aleph_\omega$ as a strong limit was shown by Gitik to be equiconsistent with the existence of a measurable $\kappa$ of Mitchell order $\kappa^{++}$.)

REPLY [7 votes]: It follows from the work of Gitik and Mitchell Indiscernible sequences for extenders, and the singular cardinal hypothesis that the hypothesis implies the existence of an inner model with overlapping extenders. 
As explained in the comments by Andres, it follows from the work of
Gitik, Schindler and Shelah Pcf theory and
Woodin cardinals that one can get $PD$ (Projective Determinacy). 
It seems that the results of the above paper are currently the best ones.