Gessel's conjecture on the two-sided Eulerian polynomial

Let $A_n(s,t)$ denote the two-sided Eulerian polynomial. For $n\geq 1$, the polynomial is considered in the basis

$$(st)^i(s+t)^j(1+st)^{n+1-j-2i},$$

with indices satisfying $0\leq i-1$, $j$, and $j+2i\leq n+1$. Gessel's conjecture. There exist nonnegative integers $\gamma_{n,i,j}$ such that

$$A_n(s,t)=\sum_{i,j}\gamma_{n,i,j}(st)^i(s+t)^j(1+st)^{n+1-j-2i}.$$

The conjecture asserts nonnegativity of the coefficients in the basis determined by the symmetries of the two-sided Eulerian polynomial. The source gives no resolution status, so it remains open here.