Asymptotic independence of descent sets and being an n-cycle

For each $n$, let $[n-1]=\{1,\ldots,n-1\}$, and let $\beta_n(I)$ denote the number of permutations in $S_n$ with descent set $I\subseteq[n-1]$, while $\beta^{\mathrm{cyc}}_n(I)$ denotes the number of $n$-cycles with descent set $I$. Asymptotic independence conjecture.

$$\lim_{n\to\infty}\max_{\varnothing \subsetneqq I \subsetneqq [n-1]} \left| \frac{n\, \beta^{\mathrm{cyc}}_n(I)}{\beta_n(I)} - 1\right| = 0.$$

This asserts that, uniformly over all non-empty proper descent sets, having descent set $I$ and being an $n$-cycle are asymptotically independent, with the proportion of $n$-cycles tending to $1/n$ as it does among all permutations. The source presents this as a conjectural strengthening of the known asymptotic independence result for alternating permutations.