Cutting-distance conjecture for even k-ary words

For a $k$-ary word $W$, let $c(W)$ be the minimum number of cuts needed to partition $W$ into consecutive blocks that can be permuted to form a shuffle square. A word is even if every letter occurs an even number of times.

Cutting-distance conjecture. For each $k\geqslant2$, every even $k$-ary word $W$ satisfies

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

The binary case is posed separately with the stronger bound $c(W)\leqslant2$, while the proposed generalization is compatible with the known ternary example having cutting distance $3$.