Prolongation intersection conjecture for claw-tree varieties

Assume the group-based model is abelian. Let $K_{n,1}$ be a claw tree with $n$ leaves, and let $X(T)$ denote the associated variety. A prolongation $T_i$ of $K_{n,1}$ is obtained by subdividing an edge so that it has exactly two inner vertices, each of valency at least three.

Prolongation intersection conjecture.

$$X(K_{n,1})=\bigcap_i X(T_i),$$

where the intersection is scheme-theoretic and ranges over all such prolongations $T_i$ of $K_{n,1}$.

This conjecture gives a geometric description of the claw-tree variety through varieties of two-inner-vertex trees. It is presented as a natural conjecture motivating the method, and no proof or resolution is supplied in the given text.