Pattern-avoidance characterization of restricted-stack-sortable permutations

Let $t\geq 1$, $n\geq 0$, and $p\in\{213,321\}$. A permutation is $t$-stack-sortable if it can be sorted using $t$ applications of the stack-sorting operator, and it avoids a pattern if it contains no subsequence order-isomorphic to that pattern. Restricted-stack characterization conjecture. The $t$-stack-sortable $p$-avoiding permutations of length $n$ are in one-to-one correspondence with the permutations of length $n$ that avoid the patterns $132$ and $12\cdots(t+2)$ simultaneously. This conjecture is motivated by computational agreement with the enumerations for $p\in\{213,321\}$ and is presented as a conjecture related to the 321-machine; its general validity remains open.