The vanishing proportion conjecture for square-free M-unambiguous ternary and larger alphabets

Let $\Sigma$ be an ordered alphabet with $|\Sigma|\ge 3$. A word $w\in\Sigma^*$ is square-free-$M$-unambiguous when it is not $M$-equivalent to any other distinct square-free word. Vanishing proportion conjecture.

$$\lim_{k\rightarrow\infty}\frac{|\{w\in\Sigma^*:w\text{ is square-free-$M$-unambiguous and }|w|=k\}|}{|\{w\in\Sigma^*:w\text{ is square-free and }|w|=k\}|}=0.$$

The conjecture is motivated by computations in the appendix showing that, up to length $60$, the proportion of square-free-$M$-unambiguous words eventually decreases steadily. Its asymptotic assertion remains open in the source.