The conjecture that P-distinguishability is always trivial for M-classes

Let $\Sigma$ be an ordered alphabet and let $w\in\Sigma^*$ be $M$-ambiguous. Write $[w]$ for the $M$-class of $w$. A class is trivially $\mathbb{P}$-distinguishable when it is a single $\mathbb{P}$-equivalence class. P-distinguishability conjecture. The class $[w]$ is $\mathbb{P}$-distinguishable if and only if $[w]$ is trivially $\mathbb{P}$-distinguishable. This conjecture proposes that, for $M$-ambiguous words, no nontrivial $\mathbb{P}$-distinguishability occurs.