The bounded-index virtually solvable fundamental group conjecture

Let $d$ and $c$ be the dimension and absolute coregularity, respectively, and let $(X,D)$ be a klt Calabi–Yau type pair of dimension $d$ and absolute coregularity $c$. Write $\pi_1^{\rm reg}(X,D)$ for its fundamental group of the regular locus. Virtually solvable fundamental group conjecture. There exists a constant $i(d,c)$ depending on the dimension and coregularity such that $\pi_1^{\rm reg}(X,D)$ is virtually solvable of length at most $2d-1$ and index at most $i(d,c)$. This conjecture seeks an effective bound on the finite index after replacing the generally unavailable virtually abelian conclusion by virtual solvability. The source explains that bounded index for abelian subgroups cannot hold in general, whereas the proposed solvable bound remains open.