Tropicalization conjecture for the type C cluster configuration space

Let $N$ be the label set for the binary geometry defining the type C cluster configuration space, and let $\mathcal{M}_{\mathrm{C}_{n-1}}$ be its very affine part in $u$-coordinates. An axially symmetric phylogenetic tree (ASPT) is an axially symmetric tree with $2n$ labeled leaves, and for each ASPT $(T,\varphi)$ let $C_{T,\varphi}$ denote the associated cone.

Tropicalization conjecture. The tropicalization $\operatorname{Trop}\mathcal{M}_{\mathrm{C}_{n-1}}$ is the space of ASPTs: the fan formed by the cones $C_{T,\varphi}$ with $(T,\varphi)$ ranging over all ASPTs.

This conjecture proposes a combinatorial description of the tropicalization of the type C cluster configuration space, identifying its fan structure with the complex of axially symmetric phylogenetic trees. The source presents this as the first conjecture; no resolution status is supplied.