Eventual constancy conjecture for 213- and 132-avoiding permutations

Let $f_n^k(\sigma)$ denote the number of permutations of length $n$ associated with the pattern $\sigma$ and fixed by the $k$th iterate of the fundamental bijection. For $\sigma\in\{213,132\}$ and $k\geq1$, the conjecture is

Eventual constancy conjecture. For each $\sigma\in\{213,132\}$ and $k\geq1$, $f_n^k(\sigma)$ is eventually constant as $n$ becomes large. The paper conjectures that for $n\geq8$ the values for $1\leq k\leq14$ are given by the displayed table: for $213$, $(2,4,7,9,2,9,8,9,8,7,2,16,2,10)$, and for $132$, $(2,3,6,7,2,7,8,7,7,5,2,13,2,9)$.

The table records computationally suggested eventual values for the first fourteen iterates; the general eventual-constancy assertion remains open in the supplied text.