Prym-Brill-Noether dimension conjecture for general k-gonal covers

Let $r \geq 1$ and $k \geq 3$. For a general étale double cover $f \colon \widetilde{C} \to C$ in the $k$-gonal locus in $\mathcal{R}_g$, the Prym-Brill-Noether variety $V^r(C,f)$ is expected to satisfy

$$\mathrm{dim} V^r(C,f) = \left\{ \begin{array}{ll} g-1- (k-1)(r+1) + {{k}\choose{2}} &\text{if } k < r+1 \\ g-1- {{r+1}\choose{2}} &\text{if } k \geq r+1. \end{array} \right.$$

This refines the preceding upper bound for general covers in the $k$-gonal locus, which is not optimal in general. The conjecture gives the expected dimension in both ranges, while the source does not establish the asserted equality in general.