The asymptotic index-subgroup count conjecture for braid groups

Let $B_n$ be the braid group on $n$ strands, and let $a_k(B_n)$ denote the number of subgroups of index $k$ in $B_n$. For $n$ sufficiently large, write $c(k/n)$ for a constant depending only on the ratio $k/n$. Asymptotic index-subgroup count conjecture. For all $n\gg 0$,

$$a_k(B_n)= \begin{cases} 1 & \text{if } n\nmid k,\\ c(k/n)\cdot n+1 & \text{if } n\mid k. \end{cases}$$

The preceding computations establish this pattern for several ranges of $k$ and $n$, including $a_k(B_n)=1$ in specified nondivisible ranges and linear formulas when $k=n$ or $k=2n$. The conjecture predicts that these behaviors persist for all sufficiently large braid groups.