Pinnacle-set refinement of gamma-positivity for inverse descent polynomials

For $n\geq1$ and $S\subseteq[n]$, let $\operatorname{Pin}(\pi)$ be the set of values at peak positions of $\pi\in\mathfrak{S}_n$, and define

$$\hat{A}_{n,S}^{\operatorname{ides}}(t)=\sum_{\substack{\pi\in\mathfrak{S}_n\\\\ \operatorname{Pin}(\pi)=S}}t^{\operatorname{ides}(\pi)+1}.$$

A polynomial is $\gamma$-positive with center of symmetry $(n+1)/2$ if its gamma-basis coefficients relative to that center are nonnegative. Pinnacle-set gamma-positivity conjecture. For all $n\geq1$ and $S\subseteq[n]$, the polynomial $\hat{A}_{n,S}^{\operatorname{ides}}(t)$ is $\gamma$-positive with center of symmetry $(n+1)/2$. This is a refinement of the peak-number version by fixing the entire pinnacle set.