DLT extension conjecture

Let $(X,\Delta)$ be a projective divisorial log terminal pair, with $\Delta$ a $\mathbb{Q}$-divisor, and write $\lfloor\Delta\rfloor=S$. Suppose that $K_X+\Delta$ is nef and

$$K_X+\Delta\sim_{\mathbb{Q}}D\geq 0,$$

where $S\subset\operatorname{Supp}D$. DLT extension conjecture. For all sufficiently divisible integers $m\geq 2$, the restriction map

$$H^0\left(X,\mathcal{O}_X(m(K_X+\Delta))\right)\to H^0\left(S,\mathcal{O}_S(m(K_X+\Delta))\right)$$

is surjective.

The conjecture is an extension statement for sections from the reduced round-down divisor. The paper notes that it holds when $K_X+\Delta$ is semi-ample and uses it in reductions concerning good minimal models.