CLRW's Prym-Brill-Noether dimension conjecture for general k-gonal curves

Let $\pi \colon \widetilde{C} \to C$ be an unramified double cover of a general $k$-gonal curve, with $k \geq 3$. Let $\ell = \left\lceil \frac{k}{2} \right\rceil$ and define

$$n(r,k)=\begin{cases}{{\ell+1}\choose{2}}+\ell(r-\ell)&\text{if }\ell\leq r-1,\\{{r+1}\choose{2}}&\text{if }\ell\geq r.\end{cases}$$

Here $V^r(C,\pi)$ denotes the Prym-Brill-Noether locus. CLRW's Prym-Brill-Noether dimension conjecture. If $g \gg n(r,k)$, then

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

The preceding results provide the upper bound $\dim V^r(C,\pi)\leq g-1-n(r,k)$, but it is not sharp in every small-genus example. The conjecture predicts that this bound is attained when the genus is sufficiently large.