Coregularity structure conjecture for regional fundamental groups

Let $X$ be a rationally connected variety of dimension $n$, regularity $r$, and coregularity $c$. Assume that $G:=\pi_1^{\rm reg}(X)$ is finite. Coregularity structure conjecture. The group $G$ admits a subnormal subseries

$$A_0\triangleleft A_1\triangleleft A_2\triangleleft G,$$

such that $A_0$ acts on either $\mathbb{D}^{r-1}$ or $S^{r-1}$, $A_1/A_0$ is an abelian group of rank at most $r$, and $A_2/A_1$ has order at most $N(c)$, where $N(c)$ depends only on the coregularity $c$.

This decomposes the finite regional fundamental group into contributions from the dual complex, elements fixing its strata, and the minimal log canonical center. The source presents the bound depending only on coregularity as conjectural and open.