The superexponential growth and exponential decay conjecture for Kazhdan–Lusztig polynomials

For each $n$, let $\mathrm{num}_{n}$ be the number of distinct polynomials among $\{\mathrm{P}_{w}\mid w\in\mathrm{S}_{n}\}$, and let $\mathrm{num}_{n}^{\%}$ be the ratio of this number to the size of the corresponding multiset. Growth and decay conjecture. The number of distinct Kazhdan–Lusztig polynomials grows superexponentially, meaning

$$\mathrm{num}_{n}\in\Omega(\gamma^{n})\quad\text{for all }\gamma\in\mathbb{R}_{>1},$$

and the percentage of distinct polynomials decays exponentially, meaning

$$\mathrm{num}_{n}^{\%}\in O(\delta^{n})\quad\text{for some }\delta\in(0,1).$$

The claim is based on computed data for $n$ through $11$ and concerns the asymptotic distribution of distinct first-row Kazhdan–Lusztig polynomials.