The extension conjecture for divisorial log terminal pairs

Let $(X,\Delta)$ be an $n$-dimensional projective divisorial log terminal pair such that $\Delta$ is a $\mathbb Q$-divisor, and write $\lfloor\Delta\rfloor=S$. Assume that $K_X+\Delta$ is nef and

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

where $S\subset\operatorname{Supp}D$. Then, for all sufficiently divisible integers $m\geq 2$, the restriction map

$$H^0\bigl(X,\mathcal O_X(m(K_X+\Delta))\bigr)\to H^0\bigl(S,\mathcal O_S(m(K_X+\Delta))\bigr)$$

is surjective.

This extension statement concerns extending pluricanonical sections from the reduced boundary to the ambient dlt pair. It is described as one of the conjectural ingredients related to abundance, and its general status is not settled in the source.