SASE equivalence conjecture for decreasing-increasing patterns

Let $M(p)$ denote the permutation matrix of a pattern $p$, and set

$$F_k=M((k-1)(k-2)\cdots1k),\qquad J_k=M(k(k-1)\cdots1).$$

Here $F_k\underset{1-\text{SASE}}{\sim}J_k$ denotes equivalence under $1$-SASE. SASE equivalence conjecture. For all $k>2$, we have $F_k\underset{1-\text{SASE}}{\sim}J_k$. This would fully extend the cited theorem to alternating and reverse alternating permutations; it generalizes earlier results, and empirical data suggests that it generates most equivalences for alternating permutations.