Asymptotic density of binary shuffle squares

Let $[k]^*$ be the set of finite words over $[k]=\{1,\ldots,k\}$, and let $\mathcal{S}_k(n)$ be the set of words in $[k]^{2n}$ that can be partitioned into two disjoint, identical subwords. A word in $\mathcal{S}_k(n)$ is a shuffle square, and $n$ is its semi-length.

Binary shuffle-square density conjecture. As $n\to\infty$, asymptotically half of all binary words of length $2n$ are shuffle squares; equivalently,

$$|\mathcal{S}_2(n)|=\left(\frac{1}{2}-o(1)\right)2^{2n}.$$

The parity condition that every letter occurs an even number of times is necessary for a word to be a shuffle square, and the conjecture asserts that almost every binary word satisfying this condition is a shuffle square. The conjecture is resolved by the paper's main theorem, which proves the stronger estimate $|\mathcal{S}_2(n)|=\left(\frac{1}{2}-o(n^{-1/15})\right)2^{2n}$.