Toric Property $N_p$ conjecture inspired by Mukai's conjecture

Let $X$ be a projective toric variety of dimension $n$, let $T$ be its algebraic torus, and let $L$ be an ample line bundle on $X$. Let $p$ be a nonnegative integer, and let Property $N_p$ denote the standard syzygetic property for the embedding defined by $L$. The toric Property $N_p$ conjecture. If

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

for every $T$-invariant curve $C$, then $L$ satisfies Property $N_p$. This extends the integral-closedness or projective-normality case $p=0$ and is motivated by Mukai's conjecture. The source presents it as a conjecture and gives no resolution; related results in the paper provide evidence in its direction.