Conjectured enumeration of 132-avoiding permutations fixed by the fundamental bijection

Let $t_n^k(\tau)$ denote the number of permutations of length $n$ avoiding the pattern $\tau$ that are fixed by the $k$th iterate of the fundamental bijection. For $ngeq2$, the conjectured values for $t_n^2(132)$ are

$$t_n^2(132)=\begin{cases} k^3+3k^2+2k-1 & n=3k,\\ k^3+4k^2+4k & n=3k+1,\\ k^3+5k^2+7k+2 & n=3k+2.\end{cases}$$

For $k\geq3$, the conjectured values of $t_n^k(132)$ are $3n-4$ when $k=3$, $2n-1$ when $k=4$, $n+2$ when $k=5$, and $5$ when $k\geq6$.

These formulas concern the remaining enumeration case after the paper establishes eventual vanishing for the pattern $123$; enumeration for $\mathcal{T}^k(132)$ with $k\geq2$ is presented as complicated and unresolved.