Urbinati's global generation conjecture for Weil divisors

Let $X$ be a complex normal projective variety and $D$ a Weil divisor on $X$. Global generation conjecture. There is an ample Cartier divisor $H$ on $X$ such that for any $m \in \mathbb N$, the sheaf $\mathscr O_X(m(D+H))$ is globally generated.

The conjecture is closely related to discreteness of jumping numbers: a proof would give an unconditional positive answer to that question. The paper establishes a weak form for isolated points of the non-$\mathbb Q$-Cartier locus of $D$.