Intersection behavior of the split-Jacobian fans

Characterize the intersection behavior in the tropical moduli space of genus-2 curves of the fans associated with $L_k$ for all values of $k$.

Background

For each residue class $k$ relatively prime to the splitting degree $d$, the paper constructs a fan associated with the locus $L_k$ of genus-2 curves having a corresponding $d$-split Jacobian. These fans map into the tropical moduli space $M^{tr}_2$, and their images may overlap because the parameterization of split Jacobians is not presented as a set of representatives.

The paper conjectures that the fans associated with $L_1$ and $L_{d-1}$ have the same image and verifies this for $d=3$, with sporadic higher-degree examples suggesting the same phenomenon. It leaves the general intersection pattern for all $k$ unresolved.