Toric intersection conjecture for group-based tree varieties

Let $T$ be a tree in the setting of the paper's prolongation construction, and let $X(T)$ be its associated toric variety. For each relevant prolongation $T_i$, let $X(T_i)$ be the corresponding toric variety. The toric intersection of toric varieties is the closure of the distinguished torus obtained by saturating the sum of the lattices of characters that restrict trivially to the respective subtori.

Toric intersection conjecture. The toric variety $X(T)$ is the toric intersection of all the toric varieties $X(T_i)$.

This refines the preceding scheme-theoretic intersection proposal at the level of toric varieties and character lattices. The paper presents it as a conjecture in the setting of its earlier construction and does not prove it in general.