Expected dimension conjecture for Prym–Brill–Noether loci of generic Prym curves

Let $C$ be a curve of genus $g$, and let

$$f:\widetilde{C}\to C$$

be a generic Prym curve, meaning an unramified double cover. Let $V^r(C,f)$ denote the Prym–Brill–Noether locus of rank-$r$ linear series associated to $f$, and let $n(r,k)$ be the bound defined in the paper for a $k$-gonal curve. Suppose that $g\gg n(r,k)$. Expected-dimension conjecture. Then

$$\dim V^r(C,f)=g-1-n(r,k).$$

The claim predicts that the lower bound on the codimension established for the $k$-gonal Prym–Brill–Noether locus is attained for generic Prym curves when the genus is sufficiently large. The source gives no resolution of this expectation; the preceding discussion notes that the bound is not always strict and is expected to be strict for sufficiently high genus.