The splitting type stratification conjecture for Brill–Noether loci

Let $C$ be a general curve of genus $g$ and gonality $k\geq 2$, and let $\pi:C\to\mathbb{P}^1$ be a degree-$k$ morphism. For a line bundle $L$ on $C$, write

$$\pi_*L\cong\mathcal{O}(\bm{\mu})$$

for its splitting type, and let $W^{\bm{\mu}}(C)$ denote the corresponding locally closed stratum. The partial order $\leq$ and magnitude $\lvert\bm{\mu}\rvert$ are as defined for splitting types.

The splitting type stratification conjecture.

1. $W^{\bm{\mu}}(C)$ is contained in the closure of $W^{\bm{\lambda}}(C)$ if and only if $\bm{\mu}\leq\bm{\lambda}$.
2. $W^{\bm{\mu}}(C)$ is smooth.
3. $W^{\bm{\mu}}(C)$ has dimension $g-\lvert\bm{\mu}\rvert$ if $g\geq\lvert\bm{\mu}\rvert$, and is empty otherwise.
4. $W^{\bm{\mu}}(C)$ is irreducible if $g>\lvert\bm{\mu}\rvert$.

These assertions predict the closure relations, smoothness, dimensions, and irreducibility of all splitting-type strata in $W^r_d(C)$. The source presents them as a conjectural description for a general curve of fixed gonality; no resolution is supplied in the provided text.