The interpretation conjecture for the action of profinite words on subsets

Let $M$ be the finite monoid under consideration, let $X$ be its generating alphabet, let $R$ be the relevant set on which $Omega(X)$ acts, and let $P(M)$ denote the power set of $M$. For $\tau\in\Omega(X)$, define its interpretation $\tau\Lambda\in P(M)$ recursively by the interpretation map described in the surrounding \text. If $Y\subseteq R$, write $Y\cdot\tau$ for the action of $\tau$ on $Y$. Interpretation conjecture. For every $\tau\in\Omega(X)$ and every $Y\subseteq R$,

$$Y\cdot\tau=Y(\tau\Lambda).$$

This conjecture proposes that the action of a profinite word on a subset of $R$ is exactly captured by multiplying the subset by its interpretation in the power-set monoid $P(M)$.