Asymptotic equivalence conjecture for powerful 132-avoidance

Let $\Omega_n^T(\tau)$ be the set of permutations in $S_n$ whose powers indexed by $T$ avoid $\tau$, and let $\operatorname{PAv}_n(132)$ and $\operatorname{SAv}_n(132)$ denote the sets of permutations that powerfully and strongly avoid $132$, respectively.

Powerful 132-avoidance asymptotic conjecture. As $n$ tends to infinity,

$$\left|\Omega_n^{\{1,2,3\}}(132)\right|\sim\left|\operatorname{PAv}_n(132)\right|\sim\left|\operatorname{SAv}_{n-1}(132)\right|.$$

The conjecture is based on numerical evidence. The source explicitly says that none of the three implied asymptotic equivalences is known.