The Minimal Syzygy Conjecture for general curves

Fix integers $r,s\geq1$, set $d:=rs+r$ and $g:=rs+s$, so that $\rho(g,r,d)=0$, and let $C$ be a general curve in $\mathcal M_g$. For every $L\in W^r_d(C)$ and every integer $p$ satisfying

$$0\leq p\leq\frac{r-2s}{s+1},$$

Minimal Syzygy Conjecture. One has

$$K_{p,2}(C,L)=0.$$

This conjecture interpolates between Green's conjecture for general curves and the Maximal Rank Conjecture. The source introduces it as new and gives no resolution status.