Refined Hall–Littlewood filling identity conjecture

Let $\mathrm{FIL}(\lambda,\tau,C)$ be the set of all fillings with shape $\lambda$, big basement $\tau$, and column sets $C$. Let $w_0$ be the decreasing basement permutation, and let $\tau\lambda$ and $\tau w_0$ denote the corresponding permuted shape and basement. Then

Refined Hall–Littlewood filling identity conjecture.

$$\sum_{F\in\mathrm{FIL}(\lambda,w_0,C)}q^{\operatorname{maj}(F)}t^{\operatorname{inv}(F)} = \sum_{F\in\mathrm{FIL}(\tau\lambda,\tau w_0,C)}q^{\operatorname{maj}(F)}t^{\operatorname{inv}(F)}.$$

This refinement is motivated by the known equality for modified Hall–Littlewood polynomials and by the authors' bijective proof of its $t=0$ case; the full equality is presented as a computer-supported conjecture.