Uniform admissible-section conjecture for disconnected klt Calabi–Yau pairs

For a positive integer $n$ and a natural number $d$, let $A(X,D)$ denote the space of admissible sections associated with the divisor $D$. Uniform admissible-section conjecture. If $(X,B)$ is a possibly disconnected klt pair of dimension at most $d$ and

$$n(K_X+B)\sim 0,$$

then there exists a constant $N(n,d)>0$ such that $A(X,N(K_X+B))$ contains a nonzero admissible section. The conjecture is proved in the paper when $X$ has dimension $1$; it remains open in the stated generality.