Weighted-network formula for tropical determinantal valuations

Let $x_1,x_2,x_3$ be points in the affine Grassmannian, and let $\omega_i,\omega_j,\omega_k$ be fundamental weights of $SL_n$ with $i+j+k=n$. For points $p,q$ in the affine Grassmannian, let $d(p,q)$ be the coweight-valued distance, and let $f_{ijk}^t$ denote the tropicalized canonical function. Weighted-network conjecture. For a positive configuration of points, one has

$$f_{ijk}^t(x_1,x_2,x_3)=\min_p\left\{\omega_i\cdot d(p,x_1)+\omega_j\cdot d(p,x_2)+\omega_k\cdot d(p,x_3)\right\},$$

where the minimum is over all $p$ in the affine Grassmannian. This is the weak form; the strong form asserts the same formula for any configuration of points. The formula would give a geometric interpretation of the tropicalized canonical functions and hence of intersection pairings between higher laminations. The paper notes that one inequality is immediate, that the conjecture holds when one of $i,j,k$ is zero, and that it is true for $SL_2$ and asymptotically; the general assertion remains open.