Ito–Lozovanu conjecture on syzygies of polarized abelian varieties

Let $(A,L)$ be a polarized abelian variety of dimension $g$, and let $p$ be a positive integer. Define the $\mathbb Q$-ample divisor

$$D=\frac{1}{p+2}L.$$

Assume that for every abelian subvariety $B$ of $A$,

$$(D^{\dim B}\cdot B)>(\dim B)^{\dim B}.$$

Ito–Lozovanu conjecture. Under these assumptions, $L$ satisfies property $(N_p)$, meaning that the first $p$ steps of the minimal graded free resolution of its section algebra are linear. This conjecture gives numerical conditions on abelian subvarieties that are expected to imply higher syzygy properties; the source attributes it to Ito and Lozovanu and treats it as open.