Ilinca–Kahn's precise asymptotics conjecture for maximal independent sets in B(n,k)

For positive integers $n$ and $k\leq n$, let $B(n,k)$ be the bipartite graph with vertex set consisting of the $k$-subsets and $(k-1)$-subsets of $[n]$, with adjacency given by inclusion. Write $\operatorname{mis}(B(n,k))$ for the number of maximal independent sets of this graph.

Ilinca–Kahn's conjecture. The precise asymptotics satisfy

$$\operatorname{mis}(B(n,k))=(1+o(1))\cdot n\cdot 2^{\binom{n-1}{k-1}}.$$

This conjecture strengthens the known logarithmic asymptotics $\log_2\operatorname{mis}(B(n,k))=(1+o(1))\binom{n-1}{k-1}$ proved by Ilinca and Kahn. The source presents the precise asymptotic formula as an open question.