Wang's polynomial-growth conjecture for fundamental groups of klt pairs

Let $(X,\Delta)$ be a projective klt pair with nef anti-log canonical divisor $-(K_X+\Delta)$. Here $X_{\operatorname{reg}}$ denotes the regular locus of $X$, and $\pi_1(X_{\operatorname{reg}})$ its fundamental group. Wang's conjecture. The group $\pi_1(X_{\operatorname{reg}})$ has polynomial growth. This conjecture would provide the fundamental-group input for a uniformization theorem and is described as difficult in general; the paper explains that its proof is not needed for the new structure theorem.