Major-index Wilf classes are generated by inversion

Let $M\mathcal I_n(\pi;q)$ denote the major-index generating function over involutions in $\mathfrak S_n$ avoiding a pattern $\pi$, and let $[\pi]_{M\mathcal I}$ denote the corresponding $M\mathcal I$-Wilf equivalence class. Let $r_1(\pi)=\pi^{-1}$. A permutation is an involution when it equals its inverse. Major-index Wilf-class conjecture. The only non-singleton $M\mathcal I$-Wilf equivalence classes are

$$[\pi]_{M\mathcal I}=\{\pi,r_1(\pi)\}$$

when $\pi$ is not an involution. The claim predicts that, apart from a pattern and its inverse, no distinct patterns are $M\mathcal I$-Wilf equivalent. The paper notes that this is supported by computational data, while the length-three case is proved.