Albert's conjecture on pattern-avoiding subsequence expectations

Let $X$ be a finite proper subset of $\bigcup_{j\geq 2}\mathfrak{S}_j$, meaning that $X$ does not contain both an identity permutation $12\cdots j$ and a reverse identity permutation $k\cdots 21$. Let $s_n(X)$ be the number of permutations in $\mathfrak{S}_n$ avoiding every pattern in $X$, and let $E_X(n)$ be the expected length of the longest subsequence avoiding every pattern in $X$ in a uniformly random permutation of $\mathfrak{S}_n$. Albert's conjecture. If $X$ is proper, then

$$\lim_{n\to\infty}s_n(X)^{1/n}=\frac{1}{4}\left(\lim_{n\to\infty}\frac{E_X(n)}{\sqrt{n}}\right)^2.$$

The limit on the left exists, whereas the existence of the limit on the right is part of the conjecture. The result would relate the exponential growth rate of a pattern-avoidance class to the asymptotic scale of its longest avoiding subsequences.