Equality of the images of L₁ and Lᵈ⁻¹

Prove or disprove that the fans associated with L₁ and L_{d−1} have the same image in the moduli space M₂ᵗʳ of tropical curves.

Background

The locus 𝕋𝓛d of genus-2 tropical curves with d-split Jacobians is decomposed into subsets L_k indexed by units k modulo d. Each subset is organized by a fan whose cones are mapped into the tropical moduli space M₂ᵗʳ. The paper verifies equality of the images for L₁ and L{d−1} when d = 3 and reports sporadic evidence in higher degrees, but does not establish the general claim.

The conjecture asks whether this apparent symmetry persists for every d, which would clarify redundancy in the parameterization of split Jacobians and the structure of the corresponding tropical moduli locus.