The number of maximal independent sets in the Hamming cube

Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically $$2n2^{N/4},$$ as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and R\"odl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them "stability" results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$.