Ein–Lazarsfeld–Mustopa syzygy bundle stability conjecture

Let $X$ be a smooth projective variety, and let $A$ and $P$ be line bundles on $X$. Assume that $A$ is very ample, and set $L_d:=dA+P$ for every positive integer $d$. Ein–Lazarsfeld–Mustopa's conjecture. For $d\gg0$, the syzygy bundle $S_{L_d}$ is $A$-stable. This conjecture concerns asymptotic stability of syzygy bundles; the surrounding discussion states that the corresponding question is open in general, with complete answers known in low dimensions such as curves and surfaces.