Almost all permutations are Kostant negative

Let $S_n$ be the symmetric group, and let $obreak\mathbf{p}_n$ denote the number of Kostant positive elements in $S_n$, where an element is Kostant positive if Kostant's problem for the corresponding simple highest weight module in the principal block of category $\mathcal{O}$ for $\mathfrak{sl}_n(\mathbb{C})$ has a positive answer. Almost-all-permutations conjecture. Almost all elements in $S_n$ are Kostant negative, in the sense that

$$\frac{\mathbf{p}_n}{n!}\to 0$$

as $n\to\infty$. This conjecture concerns the asymptotic prevalence of negative answers to Kostant's problem among simple highest weight modules in the principal block; the paper proves this assertion.