Dokos et al.'s major-index distribution conjecture for three length-four patterns

Let $\mathcal{S}_n$ be the set of permutations of $[n]$, and let $\mathcal{S}_n(\tau)$ denote the permutations avoiding the pattern $\tau$. For a permutation $\pi$, its major index is

$$maj(\pi)=\sum_{i\in\mathcal{D}(\pi)},$$

where $\mathcal{D}(\pi)=\{i\mid \pi_i>\pi_{i+1}\}$. Dokos et al.'s conjecture. The major index is equally distributed on the sets $\mathcal{S}_n(2413)$, $\mathcal{S}_n(1423)$, and $\mathcal{S}_n(2314)$. This is a refinement of Wilf-equivalence for permutation patterns; the source identifies it as a conjecture about patterns of length four or greater, without giving a resolution.