Gamma-positivity conjectures for two-sided Eulerian polynomials of classical Weyl groups

For $n\equiv 0,1\pmod 4$, let $\mathcal{A}_n$ be the alternating subgroup of $\mathfrak{S}_n$, and define

$$\mathrm{TSA}_n^+(s,t)=\sum_{\pi\in\mathcal{A}_n}t^{\mathsf{des}(\pi)+1}s^{\mathsf{ides}(\pi)+1},\qquad \mathrm{TSA}_n^-(s,t)=\sum_{\pi\in\mathfrak{S}_n\setminus\mathcal{A}_n}t^{\mathsf{des}(\pi)+1}s^{\mathsf{ides}(\pi)+1}.$$

For even $n$, define the type-$B$ polynomials $\mathrm{TSB}_n^+(s,t)$ and $\mathrm{TSB}_n^-(s,t)$ by summing $t^{\mathsf{des}_B(\pi)}s^{\mathsf{ides}_B(\pi)}$ over $\mathfrak{B}_n^+$ and $\mathfrak{B}_n^-$. For positive $n$, define the type-$D$ polynomials $\mathrm{TSD}_n^+(s,t)$ and $\mathrm{TSD}_n^-(s,t)$ analogously over $\mathfrak{D}_n^+$ and $\mathfrak{D}_n^-$. Gamma-positivity conjectures. (1) If $n\equiv0,1\pmod4$, then

$$\mathrm{TSA}_n^\pm(s,t)=\sum_{i,j}\gamma_{n,i,j}^\pm(st)^i(s+t)^j(1+st)^{n+1-2i-j},\qquad \gamma_{n,i,j}^\pm\geq0.$$

(2) If $n\equiv0\pmod2$, then

$$\mathrm{TSB}_n^\pm(s,t)=\sum_{i,j}\gamma_{n,i,j}^{B,\pm}(st)^i(s+t)^j(1+st)^{n-2i-j},\qquad \gamma_{n,i,j}^{B,\pm}\geq0.$$

(3) For every positive integer $n$,

$$\mathrm{TSD}_n^\pm(s,t)=\sum_{i,j}\gamma_{n,i,j}^{D,\pm}(st)^i(s+t)^j(1+st)^{n-2i-j},\qquad \gamma_{n,i,j}^{D,\pm}\geq0.$$

These conjectures extend gamma positivity from the two-sided Eulerian polynomial to positive and negative elements in types $A$, $B$, and $D$. The supplied text gives no resolution status for them.