A Solovay-like model at $\aleph_\omega$

Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_\omega$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_\omega))$ satisfies the following properties: (1) Every set $A\subseteq (\aleph_\omega)^\omega$ has the $\aleph_\omega$-PSP. (2) There is no scale at $\aleph_\omega$. (3) The Singular Cardinal Hypothesis (SCH) fails at $\aleph_\omega$. (4) Shelah's Approachability property (AP) fails at $\aleph_\omega$. (5) The Tree Property (TP) holds at $\aleph_{\omega+1}$. The above provides the first example of a Solovay-type model at the level of the first singular cardinal, $\aleph_\omega$. Our model also answers, in the context of ZF+$\mathrm{DC}_{\aleph_\omega}$, a well-known question by Woodin on the relationship between the SCH and the AP at $\aleph_\omega$.