Karzhemanov's base point free conjecture for weak log Fano pairs

Let $X$ be a projective algebraic variety of dimension at least $2$ over $\mathbb{C}$, and let $D$ be a $\mathbb{Q}$-boundary such that $(X,D)$ is log canonical and $-(K_X+D)$ is nef and big. Write $\llcorner D\lrcorner$ for the reduced integral part of $D$.

Karzhemanov's conjecture. Suppose that $X$ is $\mathbb{Q}$-factorial and

$$(K_X+D)\cdot S^{\dim X-1}\geqslant 0$$

for every irreducible component $S\subseteq\llcorner D\lrcorner$. Then the linear system

$$|-n(K_X+D)|$$

is free for $n\gg 0$.

The conjecture strengthens the paper's theorem by proposing base point freeness under a numerical condition on every component of the reduced boundary. The source presents it as a suggested correction to an earlier result; no resolution is supplied in the provided text.