Zhang–Kitaev equinumerosity conjecture for 213- and 321-avoiding permutations

Let $\mathfrak{S}_n^t$ be the set of permutations of length $n$ that become the identity after $t$ applications of the stack-sorting operation, and let $\mathfrak{S}_n^t(p)$ denote those that avoid the pattern $p$. Zhang–Kitaev's conjecture. For every $t\geq 1$,

$$|\mathfrak{S}_n^t(321)|=|\mathfrak{S}_n^t(213)|.$$

The paper constructs a bijection proving this equality, so the conjecture is solved.