Enumerative recurrence conjecture for generalized stack-sortable permutations

For positive integers $t$ and $n$, let $\operatorname{Sort}_{t,n}(123,132)$ be the set of length-$n$ permutations that are $t$-stack-sortable under $s_{123,132}$. Enumerative recurrence conjecture. For any positive integer $t$ and $n\geq2t+1$,

• if $n$ is odd, then

$$\lvert\operatorname{Sort}_{t,n}(123,132)\rvert=\frac{n+3}{2}\lvert\operatorname{Sort}_{t,n-2}(123,132)\rvert;$$

• if $n$ is even, then

$$\lvert\operatorname{Sort}_{t,n}(123,132)\rvert=\frac{n+4}{2}\lvert\operatorname{Sort}_{t,n-2}(123,132)\rvert.$$

This predicts an explicit length recurrence for the enumerator of $t$-stack-sortable permutations under the specified generalized stack-sorting map; the source provides no resolution status.