The tropical rank conjecture for binary phylogenetic tree models

Let $\tau$ be a binary tree on binary random variables, let $f$ be the associated multilinear map, and let $I_f$ be its prime ideal. For a $2 \times \cdots \times 2$ table, flattening along a split of the leaves means arranging its entries as a matrix whose rows and columns are indexed by the assignments on the two parts of the split. The tropical rank conjecture. The map $f$ is surjectively positive for $l=2$, and the tropical variety, respectively the positive tropical variety, of $I_f$ coincides with the set of all $2 \times 2 \times \cdots \times 2$ tables $(u_{\sigma_1 \cdots \sigma_n})$ whose flattenings along the splits of $\tau$ have tropical rank, respectively Barvinok rank, at most $2$. The claim is presented as a tropical extension of the determinantal description of the phylogenetic model; the source does not provide a resolution.