An involution-restricted major-index symmetry for layered pattern pairs

Let $\mathfrak i_j$ denote the increasing permutation of length $j$, let $\mathfrak d_j$ denote the decreasing permutation of length $j$, and let $12[\alpha,\beta]$ denote the direct sum of permutations. For integers $m$ and $k\in\{0,\ldots,m\}$, define

$$\pi_1=12[\mathfrak i_k,\mathfrak d_{m-k}],\qquad \pi_2=12[\mathfrak d_{k+1},\mathfrak i_{m-k-1}].$$

Let $M\mathcal I_n(\pi;q)$ be the major-index generating function over involutions in $\mathfrak S_n$ avoiding $\pi$. Involution-restricted layered-pair symmetry conjecture. For $n\geq0$,

$$M\mathcal I_n(\pi_1)=q^{\binom n2}M\mathcal I_n(\pi_2;q^{-1}).$$

The symmetry is computationally confirmed for all $m,n\leq9$ and $k\in\{0,\ldots,m\}$, and the paper reports that it does not appear to hold for other pattern pairs. The conjecture implies a further conjecture attributed to Dokos et al.; the case $k=1$ was proved by Yan, Ge, and Zhang.