Song-Zhu conjecture on Property N_p for projective toric varieties

Let $X$ be a projective toric variety of dimension $n\geq 2$, let $L$ be a line bundle on $X$, and let $p\geq 0$. Property $N_p$ is the syzygy property defined by the minimal graded free resolution of the section ring of $L$. Song-Zhu conjecture. If

$$L\cdot C\geq n-1+p$$

for every $T$-invariant curve $C\subset X$, then $L$ satisfies Property $N_p$. The conjecture proposes an effective intersection-theoretic criterion for syzygies of line bundles on projective toric varieties; the source states that it is motivated by Mukai's conjecture and remains open.