Vanishing probability of a Latin-square autotopism component

For $n>0$, let $S_n$ be the symmetric group on $n$ symbols, let $\mathrm{Atp}(n)$ denote the autotopism group of Latin squares of order $n$, and let $\mathbb{P}(n)$ be the probability that a randomly chosen $\alpha\in S_n$ is a component of some isotopism $(\alpha,\beta,\gamma)\in\mathrm{Atp}(n)$. Vanishing-probability conjecture.

$$\lim_{n\to\infty} \mathbb{P}(n)=0.$$

This conjecture concerns the asymptotic rarity of permutations that occur as components of autotopisms of Latin squares, complementing the known result that a random Latin square asymptotically almost surely has no nontrivial autotopism. It proposes that the corresponding probability for a random permutation also tends to zero; no resolution is supplied here.