Boissy–Lanneau's equivalence conjecture for extended Rauzy classes

Let $cpi$ be an irreducible generalized permutation. Its extended Rauzy class $cmathfrak{R}_{ex}(cpi)$ is the intersection of a minimal set containing $cpi$ and invariant under operations $a$, $b$, and $c$ with the set of irreducible generalized permutations. Define $cmathfrak{R}'_{ex}(cpi)$ instead as the minimal set containing $cpi$ and invariant under Rauzy operations $a$, $b$, and $c$, where $c(cpi)$ is undefined whenever it is not irreducible.

Boissy–Lanneau's equivalence conjecture. The sets $cmathfrak{R}_{ex}(cpi)$ and $cmathfrak{R}'_{ex}(cpi)$ coincide.

The conjecture concerns two formulations of extended Rauzy classes for irreducible generalized permutations. The source notes that C. Boissy had announced a proof while the paper was in press, but the supplied status is unknown.