Two-sided alternating gamma-expansion conjecture

Let $\mathfrak{S}_n$ be the symmetric group, and define the two-sided alternating Eulerian polynomial by

$$\widetilde{A}_n(s,t):=\sum_{\pi\in\mathfrak{S}_n}s^{\operatorname{altdes}(\pi^{-1})}t^{\operatorname{altdes}(\pi)}.$$

Two-sided alternating gamma-expansion conjecture. For every $n\geq 1$,

$$\widetilde{A}_n(s,t)=\sum_{i,j\geq 0\atop j+2i\leq n-1}\widehat{\gamma}_{n,i,j}(-st)^i(1+st)^j(s+t)^{n-1-j-2i},$$

where $\widehat{\gamma}_{n,i,j}$ are nonnegative integers. This conjecture refines the gamma-positivity phenomenon for alternating Eulerian polynomials by incorporating descents of both a permutation and its inverse; no proof or resolution is supplied here.