Asymptotically half of binary words are shuffle squares

A binary shuffle square is a binary word of even length that can be partitioned into two disjoint, identical subwords. Huang, Nam, Thaper, and the first author conjectured that as $n\to \mathrm{\infty }$, asymptotically half of all binary words of length $2n$ are shuffle squares. We prove this conjecture in a strong form, by showing that the number of binary shuffle squares of length $2n$ is $(\frac{1}{2}-o(n^{-1\mathrm{/}15}))2^{2n}$.