Almost all involutions are Kostant negative

Let $S_n$ be the symmetric group, let $\mathbf{i}_n$ denote the number of involutions in $S_n$, and let $\mathbf{pi}_n$ denote the number of Kostant positive involutions in $S_n$. Almost-all-involutions conjecture. Almost all involutions in $nS_n$ are Kostant negative, in the sense that

$$\frac{\mathbf{pi}_n}{\mathbf{i}_n}\to 0$$

as $n\to\infty$. This is the analogous asymptotic question for involutions, which represent the Kazhdan–Lusztig left cells relevant to Kostant's problem; the supplied text gives no evidence that this assertion has been resolved.