Log Abundance conjecture for klt pairs

Let $(X,\Delta)$ be a projective klt pair of dimension $n$. Let $\operatorname{Nef}(X)$ be the nef cone, let $\operatorname{Sem}(X)$ be the cone of semiample divisors, and let $\operatorname{Mov}^{n-1}(X)=\operatorname{Sem}(X)$ and $\overline{\operatorname{Mov}}^{n-1}(X)=\operatorname{Nef}(X)$. Log Abundance conjecture. If

$$K_X+\Delta\in\operatorname{Nef}(X)=\overline{\operatorname{Mov}}^{n-1}(X),$$

then

$$K_X+\Delta\in\operatorname{Sem}(X)=\operatorname{Mov}^{n-1}(X).$$

This is known for varieties of dimension at most $3$ and when $K_X+\Delta$ is big, but remains widely open in general.