Green's conjecture on graded Betti numbers of curves

Let $X$ be a curve of genus $g$. For a divisor $D$ on $X$ of degree $d$, let $g_d^r$ denote a subspace of the Riemann–Roch space $L(D)$ of dimension $r+1$, and let $\beta_{p,2}$ be the corresponding graded Betti number. Green's conjecture. One has $\beta_{p,2}\neq 0$ if and only if there exists a divisor $D$ on $X$ of degree $d$ such that a subspace $g_d^r$ of $L(D)$ satisfies $d\leq g-1$, $r=\ell(D)-1\geq 1$, and $d-2r\leq p$. This conjecture relates the graded Betti numbers of a curve to the existence of linear series such as a $g_4^1$, and is used in studying the complex gonality of modular curves. The source does not provide enough information here to determine whether the conjecture is open or resolved.