Conjecture that the Schwarz genus of the flex-point covering is 8

Let

$$\mathcal{X}:=\mathbb{CP}^9-\{\text{singular curves}\}$$

be the space of nonsingular plane cubic curves, and let

$$\widetilde{\mathcal{X}}_{\mathrm{flex}}:=\{(F,p)\in\mathcal{X}\times\mathbb{CP}^2\mid p\text{ is a flex point on the curve defined by }F\}$$

be the space of a cubic together with a flex point. The projection $(F,p)\mapsto F$ is a 9-fold covering, whose Schwarz genus is denoted by $g(\widetilde{\mathcal{X}}_{\mathrm{flex}}/\mathcal{X})$. The Schwarz genus conjecture.

$$g(\widetilde{\mathcal{X}}_{\mathrm{flex}}/\mathcal{X})=8.$$

The paper proves the bounds $8\leq g(\widetilde{\mathcal{X}}_{\mathrm{flex}}/\mathcal{X})\leq 9$; the value 8 is equivalent to the vanishing of a particular cohomology class $\mathfrak{o}_8$, and the conjecture remains open. The authors also prove that $3\mathfrak{o}_8=0$.