The eigenvector-map conjecture for generic ultra-discrete Toda fibers

Let $\mathcal{T}$ be the phase space of the ultra-discrete $(g+1)$-periodic Toda lattice, let $\mathcal{C}$ be the moduli space of compact tropical curves $\Gamma_C$, and let

$$\mathcal{T}_{C}=\Phi^{-1}(C)$$

be the fiber of the eigenvector map $\Phi:\mathcal{T}\to\mathcal{C}$. For a generic $C=(C_{-1},\cdots,C_{g-1},C_g=0)\in\mathcal{C}$ satisfying the condition referenced in the source, the eigenvector-map conjecture.

$$\mathcal{T}_{C}\simeq J(\Gamma_C).$$

If $C\in\mathbb{Z}^{g+2}$, the induced isomorphism bijects the lattice points $(\mathcal{T}_{C})_{\mathbb{Z}}$ and $J_{\mathbb{Z}}(\Gamma_C)$, and hence

$$\left|(\mathcal{T}_{C})_{\mathbb{Z}}\right|=\det\Lambda.$$

This identifies generic invariant fibers of the ultra-discrete Toda lattice with tropical Jacobians and, in the integral case, relates their lattice states to the Jacobian's lattice points. The supplied text gives no evidence that the assertion has been proved or disproved.