Row-equivalency class equality for queue inversion and HHL statistics

Let $\tau$ be a filling and let $[\tau]$ denote its row-equivalency class: the set of fillings obtained by independently permuting the entries in each row of $\tau$. For a filling $\rho$, let $\operatorname{quinv}(\rho)$ and $\operatorname{inv}(\rho)$ denote its queue inversion and HHL inversion statistics, respectively, and let $\operatorname{maj}(\rho)$ denote its major index.

Row-equivalency class conjecture. For every row-equivalency class $[\tau]$,

$$\sum_{\sigma\in[\tau]} t^{\operatorname{quinv}(\sigma)}q^{\operatorname{maj}(\sigma)} = \sum_{\sigma\in[\tau]} t^{\operatorname{inv}(\sigma)}q^{\operatorname{maj}(\sigma)}.$$

This conjecture would give a bijective explanation for the equality between the tableau formulas for modified Macdonald polynomials using queue inversion weights and HHL weights. The paper presents it as a natural question, and the supplied text gives no evidence that it has been resolved.