Symmetric joint distribution of inversion and major-index statistics on colored derangements

Let $G_{c,n}$ be the group of colored derangements, and let $\widetilde{inv}$ and $fmaj$ denote the statistics on $G_{c,n}$. For indeterminates $q$ and $t$, Symmetry conjecture. The statistic $\widetilde{inv}$ has a symmetric joint distribution with $fmaj$ over $G_{c,n}$; equivalently,

$$\sum_{\pi \in G_{c,n}}t^{\widetilde{inv}(\pi)}q^{fmaj(\pi)}=\sum_{\pi \in G_{c,n}}q^{\widetilde{inv}(\pi)}t^{fmaj(\pi)}.$$

The conjecture is computationally confirmed for every group $G_{c,n}$ with $1 \leq c \leq 7$ and $n=2$, as well as for $G_{3,3}$ and $G_{4,3}$; its general validity remains open.