Intersection behavior of the fans associated to L_k

Determine the intersection behavior in M^{tr}_2 of the fans associated with L_k for all k.

Background

The sets L_k provide the decomposition of the locus of genus-2 tropical curves with d-split Jacobians, and their associated fans encode the subdivisions induced by the reduction data. Understanding how the images of these fans intersect in M{tr}_2 would clarify the global combinatorial structure of the split-Jacobian locus and quantify the redundancy in the paper's set-theoretic description of the locus of split principally polarized tropical abelian varieties.

References

(Can we describe)/(What is) the intersection behavior in $M{tr}_2$ of the fans associated to $L_k$ for all $k$?

— Tropical split Jacobians of curves of genus 2 II  (2502.05624 - Cobigo, 8 Feb 2025) in Question immediately following the conjecture in Section 6, 'Moduli space perspective' (subsection 'Further Questions')