The asymptotic unimodality conjecture for Kazhdan–Lusztig polynomials

For each $n$, let $\mathrm{uni}_{n}$ be the proportion of distinct first-row Kazhdan–Lusztig polynomials for $\mathrm{S}_{n}$ that are unimodal, and let $\mathrm{uni}_{n}^{mult}$ be the corresponding proportion in the multiset of polynomials indexed by $\mathrm{S}_{n}$. Asymptotic unimodality conjecture. Almost all Kazhdan–Lusztig polynomials are unimodal:

$$\lim_{n\to\infty}\mathrm{uni}_{n}=\lim_{n\to\infty}\mathrm{uni}_{n}^{mult}=1.$$

The conjecture is motivated by computations and contrasts with the rapidly vanishing proportions of bimodal and trimodal examples observed in the data.