Lin–Kim's descent-statistic equivalence conjecture for six (4,4)-avoidance classes

Let $(\nu,\mu)$ be a pair of patterns of length four. For a permutation $\pi$, let $\operatorname{asc}(\pi)$ denote its number of ascents and let $\operatorname{last}(\pi)$ denote its last entry. For an inversion sequence $e$, let $\operatorname{dist}(e)$ denote the number of distinct positive entries and let $\operatorname{last}(e)$ denote its last entry. Let $I_n(\geq,-,>)$ be the inversion sequences avoiding the stated inversion-sequence pattern, and let $\mathcal{S}_n(\nu,\mu)$ be the permutations avoiding both patterns. Lin–Kim's conjecture. For every $n$ and each pair $(\nu,\mu)$ in

$$\{(4231,3241),(4231,2431),(4231,3421),(2431,3241),(3421,2431),(3421,3241)\},$$ $$\sum_{\pi \in \mathcal{S}_n(\nu,\mu)}q^{\operatorname{asc}(\pi)}v^{\operatorname{last}(\pi)}=\sum_{e \in I_n(\geq,-,>)}q^{\operatorname{dist}(e)}v^{\operatorname{last}(e)+1}.$$

This conjecture proposes a common bivariate generating-function equivalence between six permutation pattern-avoidance classes and the corresponding inversion-sequence class, refining the underlying enumeration by ascent and last-entry statistics. Its resolution is not specified in the supplied text.