Typical binary words with even weight are shuffle squares

Let $n\ge 1$, and let $s$ be chosen uniformly from the binary words $\{0,1\}^{2n}$ having an even number of ones. A typical-word shuffle-square conjecture. With high probability as $n\to\infty$, the word $s$ is a shuffle square. Here, “with high probability” means with probability $1-o(1)$. This would substantially strengthen the preceding bound for the number of bits that must be removed from a typical binary word to obtain a shuffle square, and is motivated by numerical evidence; no proof or resolution is given in the source.