The shuffle anti-square existence conjecture

For a binary even word $W$ of length $2n$, let $s(W)$ be the number of circular shifts of $W$ that are shuffle squares. Let $s_{2n}$ be the minimum of $s(W)$ over all binary even words of length $2n$. A shuffle anti-square is a binary even word for which no circular shift is a shuffle square. The shuffle anti-square existence conjecture. There exists a shuffle anti-square of length $2n$ for every $n\geqslant12$. Equivalently, $s_{2n}=0$ for every $n\geqslant12$. Computational data establish this for the listed small values but do not prove the assertion for all $n\geqslant12$; the conjecture remains open.