The Minimal Resolution Conjecture for syzygies of general curves

Let $C$ be a curve of genus $g$, and let $L\in W^r_d(C)$ be a complete base-point-free linear series, with $p\geq1$. Let $K_{p,1}(C,L)$ denote the relevant Koszul cohomology group, and let

$$\rho(g,r,d)=g-(r+1)(g-d+r).$$

Minimal Resolution Conjecture. Fix integers $g,r,d,p\geq1$ such that $g-d+r\geq0$, and assume

$$r-1-\left[\frac{g-1}{2}\right]\leq p\leq d-g+1$$

and

$${r\choose p-1}\left(-\frac{d}{r}(p-1)+2d+1-g-\frac{(r+1)(r+2)}{p+1}\right)+1>\rho(g,r,d).$$

Then for a general curve $[C]\in\mathcal{M}_g$, one has $K_{p,1}(C,L)=0$ for every $L\in W^r_d(C)$. This is intended as a wide-ranging common generalization of the maximal rank conjecture and Green's conjecture; its canonical specialization was known by Voisin, while the general statement remains open.