q-gamma positivity conjecture for alternating Eulerian polynomials

Let $\mathfrak{S}_n$ be the symmetric group, and let $\operatorname{altdes}(\pi)$ and $\operatorname{altmaj}(\pi)$ denote the alternating descent number and alternating major index of $\pi\in\mathfrak{S}_n$. Define the bivariate alternating Eulerian polynomial by

$$\widehat{A}_n(t,q)=\sum_{\pi\in\mathfrak{S}_n}t^{\operatorname{altdes}(\pi)}q^{\operatorname{altmaj}(\pi)}.$$

q-gamma positivity conjecture. The polynomial $\widehat{A}_n(t,q)$ has the expansion

$$\sum_{\pi\in\mathfrak{S}_n}t^{\operatorname{altdes}(\pi)}q^{\operatorname{altmaj}(\pi)}=\sum_{k=0}^{\lfloor(n-1)/2\rfloor}\widehat{\gamma}_{n,k}(q)q^{\binom{k+1}{2}}(-t)^k\prod_{i=k+1}^{n-1-k}(1+tq^i),$$

where $\widehat{\gamma}_{n,k}(q)\in\mathbb{N}[q]$ and $(1+q)^k$ divides $\widehat{\gamma}_{n,k}(q)$. This is presented as an alternating analogue of the known $q$-$\gamma$-positivity expansion for Eulerian--Mahonian polynomials; the claimed coefficient positivity and divisibility are not established in the supplied text.