The Tschirnhausen realization conjecture for primitive covers

Let $d \geq 2$ and $g \geq 0$. For a degree-$d$ cover $C \to \mathbb{P}^1$ with scrollar invariants $(e_1,\dots,e_{d-1})$, define

$$\operatorname{\mathcal{P}}_d:= \Big\{ (\overline{e}_1,\dots,\overline{e}_{d-1}): \sum \overline{e}_k = 1,\; 0 \leq \overline{e}_1 \leq \cdots \leq \overline{e}_{d-1},\; \text{and }\overline{e}_{k+\ell} \leq \overline{e}_k + \overline{e}_\ell \Big\} \subseteq \mathbb{R}^{d-1}.$$

The Tschirnhausen realization conjecture. The tuple $(e_1,\dots,e_{d-1})$ is achieved as the scrollar invariants of a smooth irreducible primitive cover $C \to \mathbb{P}^1$ of degree $d$ and genus $g$ if and only if

$$\frac{1}{g+d-1}(e_1,\dots,e_{d-1}) \in \operatorname{\mathcal{P}}_d.$$

The conjecture gives a precise classification of the Tschirnhausen bundles arising from primitive covers; the realization problem is completely known in degrees $2$, $3$, and $4$, while the general case remains open.