Noether's maximal rank conjecture for linear series on general curves

Fix nonnegative integers $g,r,d$, let $C$ be a general curve of genus $g$, and let $V\subseteq \mathcal{L}(D)$ be a general linear series of rank $r$ and degree $d$ on $C$. For each integer $m$, consider the multiplication map

$$\mu_m:\operatorname{Sym}^m V\longrightarrow \mathcal{L}(mD).$$

Noether's maximal rank conjecture. The maps $\mu_m$ have maximal rank for all $m$; that is, each $\mu_m$ is either injective or surjective. The conjecture is a central open problem in Brill–Noether theory, although several important cases are known.