Ilinca–Kahn conjecture for maximal independent sets in the Boolean-layer graph

For positive integers $n$ and $k$, let $\mathcal{B}_{n,k}$ be the graph with vertex set $\binom{[n]}{k}\cup\binom{[n]}{k+1}$ and edges given by inclusion, and let $\operatorname{mis}(G)$ denote the number of maximal independent sets of a graph $G$. Ilinca–Kahn conjecture.

$$\operatorname{mis}(\mathcal{B}_{n,k})=(1+o(1))n2^{\binom{n-1}{k}},$$

where the $o(1)$ term tends to $0$ as $n\to\infty$. Ilinca and Kahn had proved the corresponding logarithmic asymptotic, while this conjecture predicts the sharp leading factor for the number of maximal independent sets.