The inversion-major-index equidistribution conjecture for reduced words of the longest permutation

Let $w_0^{(n)}$ be the longest permutation in $athcalS_n$ and let $\operatorname{Red}(w_0^{(n)})$ denote its reduced words. For a reduced word $\rho$, define its inversion statistic $\operatorname{inv}(\rho)$ as in the paper, and define the major index by

$$\operatorname{maj}(\rho)=\sum_{i:\,\rho_{i+1}>\rho_i}i.$$

Inversion-major-index equidistribution conjecture. For $n\geq 1$,

$$\sum_{\rho\in\operatorname{Red}(w_0^{(n)})}q^{\operatorname{inv}(\rho)}=\frac{\left[\binom{n}{2}\right]_q!}{[1]_q^{n-1}[3]_q^{n-2}[5]_q^{n-3}\cdots[2n-3]_q^1}=\sum_{\rho\in\operatorname{Red}(w_0^{(n)})}q^{\operatorname{maj}(\rho)-\binom{n}{3}}.$$

The displayed identity is a proposed $q$-analogue of the known major-index formula for reduced words of the longest permutation, relating the newly defined inversion statistic to the major index. The supplied text does not state whether this conjecture has been proved or disproved.