Gamma-nonnegativity conjecture for inversions and excedances in the symmetric group

For a permutation $\sigma\in S_n$, let $\operatorname{inv}(\sigma)$ and $\operatorname{exc}(\sigma)$ denote its inversion number and excedance number, respectively, and define

$$S_n(q,t)=\sum_{\sigma\in S_n}q^{\operatorname{inv}(\sigma)}t^{\operatorname{exc}(\sigma)}.$$

There exist polynomials $\gamma_j(q)$ with nonnegative integer coefficients such that

$$S_n(q,t/q)=\sum_{\sigma \in S_n} q^{\operatorname{inv}(\sigma)-\operatorname{exc}(\sigma)}t^{\operatorname{exc}(\sigma)} = \sum_{0\leq j \leq (n-1)/2} \gamma_j(q) t^j(1+t)^{n-1-2j}.$$

The result would extend the analogous $\mathbb{Z}_{\geq 0}[q]$ $\gamma$-nonnegativity known for permutations in the interval $[e,(12\cdots n)]$ to the entire symmetric group. Explicit examples for $n=4$ and $n=5$ support the claim, but the statement is presented as an expectation rather than an established theorem.