Virtual nilpotence of orbifold fundamental groups of log canonical Calabi–Yau pairs

Let $(X,D_X)$ be a log canonical pair with standard coefficients, where

$$D_X=\sum_i\left(1-\frac{1}{m_i}\right)D_i$$

with each $D_i$ a prime divisor and $m_i\in\mathbb{Z}_{>0}\cup\{\infty\}$. The orbifold fundamental group $\pi_1^{\operatorname{orb}}(X,D_X)$ is the quotient of $\pi_1(X^{\operatorname{reg}}\setminus D_X)$ by the normal subgroup generated by the powers $\gamma_i^{m_i}$ of small loops $\gamma_i$ around $D_i$. Virtual nilpotence conjecture. If

$$K_X+D_X\sim_{\mathbb{Q}}0,$$

then $\pi_1^{\operatorname{orb}}(X,D_X)$ is virtually nilpotent. This extends the expected finiteness and structural properties of fundamental groups for log canonical Calabi–Yau pairs; the source provides no resolution of the conjecture.