The multiplier-ideal locus conjecture for polarized abelian varieties

Let $X$ be a $g$-dimensional abelian variety and let $B$ be an ample rational divisor on $X$. The cosupport of a multiplier ideal is the locus where that ideal is nontrivial. Multiplier-ideal locus conjecture. If

$$(B^g)>g^g,$$

then there exists an effective rational divisor $D\equiv B$ such that the cosupport of the multiplier ideal $\mathscr J(X;D)$ is, set-theoretically, an abelian subvariety of $X$. Moreover, if this abelian subvariety has dimension at least one, then the pair $(X,B)$ splits into polarized abelian submanifolds. The conjecture concerns the expected special form of log-canonical loci in the high-volume range and remains open.