The asymptotic density conjecture for Kazhdan–Lusztig polynomials

Let $\mathrm{den}_{n}$ denote the density quantity defined in the paper for Kazhdan–Lusztig polynomials indexed by permutations in $\mathrm{S}_{n}$. Asymptotic density conjecture. There is some $a\in[0,1]$ such that

$$\mathrm{den}_{n}\sim n^{-(2+a)}.$$

This is proposed from computational data and is intended as a refined description of the decay of the density; no proof or resolution is given.