Asymptotic minimal-model invariants conjecture for root covers

Let $X_n \to Y_n \to (Z,D)$ be a terminal good partial resolution of singularities of the $n$-th root cover construction. Assume that $K_{Y_n}$ is nef, and let $X'_n$ be a minimal model of $X_n$. For a partition $i_1+\ldots+i_m=d$, the Chern numbers $c_{i_1}\ldots c_{i_m}(X'_n)$ and logarithmic Chern numbers $\logc_{i_1}\ldots \logc_{i_m}(Z,D)$ are compared asymptotically by

$$\frac{c_{i_1}\ldots c_{i_m}(X'_n)}{n} \approx \logc_{i_1}\ldots \logc_{i_m}(Z,D),$$

for prime numbers $n\gg 0$. The conjecture predicts that applying the minimal model program does not change the relevant normalized invariants asymptotically.