The restricted two-stack sorting characterization for first-stack avoidance of 1324

Let a permutation be sortable by two stacks in series under the restriction that the permutation on the first stack avoids the classical pattern $1324$. Let $\operatorname{Av}(P)$ denote the permutations avoiding every pattern in a set $P$, and interpret the four displayed objects as mesh patterns. Restricted-stack sorting conjecture. The set of permutations sortable under this restriction is the set of permutations avoiding the four mesh patterns displayed in the source. This gives a conjectural pattern-avoidance description for a restricted two-stack sorting class, building on prior work on stack-sorting restrictions; the source does not state that the characterization has been proved or refuted.