The pattern-pair characterization of restricted inversion-sequence interpretations

Let $S_{n,k}$ be the quantity defined earlier in the paper, let $\rho$ and $\tau$ be patterns of length $4$, and let $d53e_n(\rho,\tau)$ denote the permutations in $d53e_n$ avoiding both $\rho$ and $\tau$. Write $\operatorname{last}(\pi)$ for the last entry of a permutation. The pattern-pair characterization conjecture. For every $0\leq k<n$, the identity

$$S_{n,k}=\left|\left\{\pi\in\mathfrak{S}_n(\rho,\tau):\operatorname{last}(\pi)-1=k\right\}\right|$$

holds if and only if $(\rho,\tau)$ is one of

$$(4321,3421),(3241,2341),(2431,2341),(4231,3241),$$ $$(4231,2431),(4231,3421),(2431,3241),(3421,2431),(3421,3241).$$

The conjecture gives a complete classification of pairs of length-four patterns yielding an interpretation of $S_{n,k}$; its resolution is not supplied in the source.