Submaximal character degree ratio conjecture for symmetric groups

Let $b_1(n)$ be the largest irreducible character degree of the symmetric group $S_n$, and for each $m\geq 2$ let $b_m(n)$ be the $m$-th largest irreducible character degree of $S_n$. Submaximal character degree ratio conjecture. For any fixed integer $m\geq 2$,

$$\frac{b_m(n)}{b_1(n)}\to 1\quad\text{as }n\to\infty.$$

This conjecture asserts that symmetric groups have no single character degree that remains asymptotically separated from the other largest degrees; the paper presents experimental evidence for the analogous behavior of the second and third largest degrees, but does not provide a proof for arbitrary fixed $m$.