Binary-network formula for ternary determinantal valuations

Let $L,M,N$ be lattices in $\mathbb{C}((t))^n$, and let $i,j,k$ be nonnegative integers with $i+j+k=n$. For lattices $L_1,\ldots,L_r$, define the tropical determinantal valuation $f^t_{a_1,\ldots,a_r}(L_1,\ldots,L_r)$ by maximizing the negative valuation of the corresponding determinant, and write $f_n^t(P)$ for the unary valuation. Binary-network conjecture. One has

$$f^t_{ijk}(L,M,N)=\min_{\text{lattices }P}\left(f^t_{i,j+k}(L,P)+f^t_{j,i+k}(M,P)+f^t_{k,i+j}(N,P)-2f_n^t(P)\right).$$

Equivalently,

$$f^t_{ijk}(L,M,N)=\min_{\text{lattices }P}\left(f^t_{ijk}(L,P,P)+f^t_{ijk}(P,M,P)+f^t_{ijk}(P,P,N)-2f^t_{ijk}(P,P,P)\right).$$

This reformulates the weighted-network conjecture in lattice terms. The paper proves related results for lattices in a common apartment and establishes asymptotic cases, but the general ternary formula remains open.