Equality of the images of the $L_1$ and $L_{d-1}$ fans

Prove that the fans associated with $L_1$ and $L_{d-1}$ have the same image in the tropical moduli space $M^{tr}_2$.

Background

The locus of genus-2 curves with $d$-split Jacobian decomposes into pieces $L_k$ indexed by units modulo $d$. The corresponding fans organize the parameter space according to the reduction data of the associated quadratic forms and map into $M^{tr}_2$.

The case $d=3$ is verified in the paper, and sporadic higher-degree examples point in the same direction, but the claimed equality of images for the two complementary indices $1$ and $d-1$ is explicitly presented as a conjecture rather than established in general.