The tropical prime form conjecture

Let $\Sigma$ be a Riemann surface with period matrix $\Omega$, and let $X,Y$ be points on a tropical graph obtained as the limit of points $x_{\alpha'},y_{\alpha'}$ on a family of curves with period matrices $\Omega_{\alpha'}$. Define the tropical prime form by

$$E^{\mathrm{trop}}(X,Y):=-\lim_{\alpha'\to0}\left(\alpha'\ln\left|E(x_{\alpha'},y_{\alpha'}\mid\Omega_{\alpha'})\right|\right).$$

Tropical prime form conjecture. At any loop order, the tropical prime form equals the scalar $s$ of Dai, namely the graph distance $d_\gamma(X,Y)$ between $X$ and $Y$ along a path $\gamma$:

$$E^{\mathrm{trop}}(X,Y)=d_\gamma(X,Y).$$

This identifies the tropical limit of the string-theoretic prime form with the graph-theoretic distance entering tropical amplitudes. The source presents the statement as conjectural, and no resolution is supplied here.