Mixed-syzygy length conjecture for line bundles on the quadric surface

Let $X=\mathbb{P}^1\times\mathbb{P}^1$ and let $L=aC_0+bf$. Let $\delta(a,b)$ denote the length of the region of mixed syzygies. Mixed-syzygy length conjecture. The length is

$$\delta(a,b)=(a-1)(b-2)=h^0(K_X+L)-\operatorname{gon}_{\max}(L)+1.$$

Equivalently, the highest value $p_{\max}$ for which $L$ satisfies Property-$(N_p)$ is

$$p_{\max}(a,b)=ab+a+b-1-(a-1)(b-2)-a=2a+2b-3.$$

This conjecture is proposed for the syzygies of line bundles on $\mathbb{P}^1\times\mathbb{P}^1$, based on the stated examples and the geometry of mixed-weight syzygies.