Mixed-syzygy length conjecture for rational surfaces

Let $X$ be a rational surface and let $L$ be an ample and base point free line bundle on $X$. Suppose that $|L|$ is a linear system containing curves $C$ of genus $g\geq 1$, and write $\operatorname{gon}_{\max}(L)$ for the maximal gonality among smooth curves in $|L|$. If

$$(-K_X\cdot L)\geq \operatorname{gon}_{\max}(L)+2,$$

then $\delta(L)$ denotes the length of the region of syzygies of mixed weights. Mixed-syzygy length conjecture. One has

$$\delta(L)=h^0(K_X+L)-\operatorname{gon}_{\max}(L)+1.$$

The conjecture is motivated by examples for rational surfaces and describes the mixed-weight part of the syzygy resolution in terms of adjoint sections and maximal gonality.