The Rauzy-class decomposition conjecture for extended Rauzy classes

Let $cmathfrak{R}_{ex}$ be an extended Rauzy class representing a connected component of a stratum $cmathcal{Q}(d_1,cldots,d_m)$ or $cmathcal{H}(d_1,cldots,d_m)$, allowing $d_i=-1$ in the quadratic-differential stratum to represent a simple pole. Let $k$ be the number of pairwise distinct entries in $c{d_1,cldots,d_m}$.

Rauzy-class decomposition conjecture. The extended Rauzy class $cmathfrak{R}_{ex}$ is a disjoint union of exactly $k$ nonempty Rauzy classes.

This predicts the precise number of Rauzy classes contained in an extended Rauzy class associated with a connected component of an Abelian- or quadratic-differential stratum. The supplied text gives no evidence of a resolution.