Two-cut conjecture for even binary words

Let $c(W)$ be the minimum integer $c$ such that $W=B_1\cdots B_{c+1}$ can be partitioned into consecutive blocks and, for some permutation $\pi$ of $[c+1]$, the rearranged word $B_{\pi(1)}\cdots B_{\pi(c+1)}$ is a shuffle square. A word is even if each letter occurs an even number of times.

Two-cut conjecture. Every even binary word $W$ satisfies

$$c(W)\leqslant 2.$$

The paper notes that even a bound by an unspecified absolute constant is not known. The analogous claim for ternary words is false, since the paper gives an example with $c(W)=3$.