Stability of syzygy bundles in all dimensions

Let $X$ be a smooth projective variety of dimension $n$, let $A$ be the ample line bundle from the preceding setup, and let $M_d$ denote the syzygy bundle associated with the relevant line bundle of degree $d$. Stability conjecture. Then $M_d$ is $A$-stable for every $d\gg 0$. This conjectures that the paper's main stability result, proved under more restrictive dimensional hypotheses, extends to smooth projective varieties of arbitrary dimension.