Gamma-positivity conjecture for peak-refined inverse descent polynomials

For $n\geq1$ and $0\leq k\leq\lfloor(n-1)/2\rfloor$, define

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

A symmetric polynomial with center of symmetry $(n+1)/2$ is $\gamma$-positive when its expansion in the corresponding gamma basis has nonnegative coefficients. Gamma-positivity conjecture. The polynomials $\hat{A}_{n,k}(t)$ are $\gamma$-positive with center of symmetry $(n+1)/2$. This refines unimodality of the inverse-descent distribution among permutations with a fixed number of peaks.