Orthogonal splitting type dimension conjecture for general covers

Let $r \geq 1$ and $k \geq 3$. For a general chain

$$\widetilde{C} \xrightarrow{f} C \xrightarrow{h} \mathbb{P}^1,$$

where $f$ is an étale double cover, $C$ has genus $g$, and $h$ has degree $k$, define the orthogonal splitting type locus

$$U^{\vec{a}}(C,f,h)=\{\mathcal{L}\in P(C,f)\mid h_*f_*\mathcal{L}\otimes\mathcal{O}_{\mathbb{P}^1}(1)\cong \mathcal{O}_{\mathbb{P}^1}(\vec{a})\oplus\mathcal{O}_{\mathbb{P}^1}(-\vec{a})\}.$$

Orthogonal splitting type dimension conjecture. Its dimension is

$$\begin{aligned} \mathrm{dim}\,U^{\vec{a}}(C,f,h)&=g-1-h^1\bigl(\operatorname{End}^q(\mathcal{O}_{\mathbb{P}^1}(\vec{a})\oplus\mathcal{O}_{\mathbb{P}^1}(-\vec{a}))\bigr)\\ &=g-1-\sum_{i>j}\left(\max\{0,a_i+a_j-1\}+\max\{0,a_i-a_j-1\}\right). \end{aligned}$$

Here $\operatorname{End}^q$ is the sheaf of endomorphisms preserving the relevant orthogonal structure. This more general conjecture is stated as implying the conjectural dimension formula for $V^r(C,f)$ in the $k$-gonal setting. The source does not provide a resolution of this conjecture.