Equality of the images of the extremal split-Jacobian fans

Prove or disprove that the fans associated to \(L_1\) and \(L_{d-1}\) have the same image in the moduli space \(M^{tr}_2\) of tropical curves of genus 2.

Background

The locus $\mathbb{T}\mathcal{L}_d$ of tropical genus-2 curves with $d$-split Jacobians decomposes into components $L_k$ indexed by units $k\in\mathbb{Z}_d^*$. Each component is organized by a fan obtained from subdivisions of the positive quadrant and mapped linearly into $M^{tr}_2$.

The cases $L_1$ and $L_{d-1}$ have explicit fan descriptions, and the paper verifies that their images agree when $d=3$, with sporadic higher-degree examples pointing in the same direction. A general proof is not supplied.