Blanco–Petersen's $q$-gamma expansion conjecture for inversions and excedances

Let $D53A_n$ be the set of permutations of $[n]$, let $\operatorname{inv}\sigma$ be the number of inversions of $\sigma$, and let $\operatorname{exc}\sigma$ be the number of excedances of $\sigma$. Define

$$S_n(q,t/q):=\sum_{\sigma\in \mathfrak{S}_n}q^{\operatorname{inv}\sigma-\operatorname{exc}\sigma}t^{\operatorname{exc}\sigma}.$$

Blanco–Petersen's conjecture. There exist polynomials $\gamma_{n,k}(q)$ with nonnegative integer coefficients such that

$$S_n(q,t/q)=\sum_{0\leq k\leq (n-1)/2}\gamma_{n,k}(q)t^k(1+t)^{n-1-2k}.$$

This conjecture proposes a $q$-analogue of the gamma expansion of the Eulerian polynomial, refining the joint distribution of inversions and excedances. The paper's abstract states that the conjecture is proved in this work.