Brill-Noether theory for totally ramified covers of the projective line

Given a curve $C$ that is a degree $k$ cover $C \to \mathbb{P}^1$ totally ramified at two points $p$ and $q$, we can seek to understand the space of degree $d$ line bundles on $C$ with prescribed ramification at $p$ and $q$. The corresponding subschemes of $\text{Pic}^d(C)$ are called transmission loci and are parameterized via elements of the (extended) $k$-affine symmetric group $\widetilde{\Sigma}_k$. Transmission loci provide a refinement of the splitting loci that have recently been extensively studied for $k$-gonal curves. Pflueger has conjectured analogues of the classic Brill-Noether theorem should hold for transmission loci. In this paper we prove Pflueger's conjectures.