Solvable-rank conjecture for fundamental groups of log canonical log Calabi–Yau pairs

Let $n$ be a positive integer, and let $(X,\Delta)$ be an $n$-dimensional log canonical log Calabi–Yau pair. Write $\pi_1^{\rm orb}(X,\Delta)$ for its orbifold fundamental group. Solvable-rank conjecture. There is a constant $k(n)$ such that there is a short exact sequence

$$1\rightarrow S \rightarrow \pi_1^{\rm orb}(X,\Delta) \rightarrow N \rightarrow 1,$$

where $S$ is a solvable group of rank at most $2n$ and $N$ is a finite group of order at most $k(n)$. This conjecture concerns the expected uniform structure of orbifold fundamental groups of higher-dimensional log canonical log Calabi–Yau pairs and is motivated in the paper by earlier conjectures of Campana and Moraga; it remains open in the stated generality.