The strong maximal rank conjecture for linear series on general curves

Let $g$, $r$, and $d$ be positive integers such that $g-d+r\geq 0$ and $0\leq \rho(g,r,d)\leq r-2$. For a general curve $X$ of genus $g$, let $W^r_d(X)$ denote the variety of divisor classes of degree $d$ and rank at least $r$, and define

$$\Sigma^r_{d,m}(X)=\{D\in W^r_d(X)\mid \mu_m\text{ does not have maximal rank}\}.$$

Here $\mu_m:\operatorname{Sym}^m H^0(X,D)\to H^0(X,mD)$ is the multiplication map and $\rho(g,r,d)=g-(r+1)(g-d+r)$. The strong maximal rank conjecture. The variety $\Sigma^r_{d,m}(X)$ has expected dimension

$$\rho(g,r,d)-1-\left\lvert md-g+1-\binom{r+m}{m}\right\rvert.$$

This refines the maximal rank conjecture by predicting not only maximal rank of the multiplication map for a general linear series, but also the dimension of the locus where maximal rank fails. The source discusses the conjecture in the range $0\leq\rho(g,r,d)\leq r-2$; the status of the general prediction is not specified in the supplied text.