The M-regularity index conjecture for syzygies of abelian varieties

Let $X$ be an abelian variety and let $A$ be an ample line bundle on $X$. The M-regularity index $m(A)$ is the largest integer $l$ such that

$$A\otimes \mathfrak m_{x_1}^{k_1}\otimes\cdots\otimes\mathfrak m_{x_p}^{k_p}$$

is $M$-regular for all distinct points $x_1,\ldots,x_p\in X$ with $\sum k_i=l$. Here a line bundle satisfies property $N_p$ when its embedding has the corresponding syzygy property. The M-regularity index conjecture. If $p\ge m$ are non-negative integers and $m(A)\ge m$, then

$$A^{\otimes k}\text{ satisfies }N_p\quad\text{for every }k\ge p+3-m.$$

This proposes a general relationship between the higher-order properties of an ample line bundle and the equations and syzygies of the associated embedding. The source presents it as a natural question raised by the preceding syzygy theorem; its resolution is not specified here.