Plancherel-Hecke longest-subsequence fluctuation conjecture

Let ${\tt LIS}$ and ${\tt LDS}$ denote the longest increasing and longest decreasing subsequence statistics for a random word under the Plancherel-Hecke setting, and let $\sigma({\tt LIS})$ be the standard deviation of ${\tt LIS}$. Write $q=f(n)\in\Theta(n^{\alpha})$ and set $\alpha_{\rm critical}=\frac12$.

Fluctuation conjecture. If $0<\alpha<\frac12$, then

$$\lim_{n\to\infty}\sigma({\tt LIS})=0;$$

whereas if $\frac12<\alpha\leq1$, then

\lim_{n\to\infty}\sigma({\tt LIS})=O(n^{\frac16}\\!}).

The same statements hold with ${\tt LDS}$ in place of ${\tt LIS}$.

This conjecture concerns fluctuations across the predicted phase transition. It is motivated by the concentration of shapes in the subcritical regime and by the $n^{1/6}$ scale associated with Plancherel-type longest-subsequence fluctuations. The source provides supporting asymptotic results for expectations, but does not establish these variance bounds in the stated generality.