Conjecture on the number of projection cones defining maximal cells in equidistant tree space

Let $\mathcal{Q}_n$ be the space whose maximal cells arise from projection cones associated with equidistant trees, and let a maximal cell mean a maximal-dimensional cell of $\mathcal{Q}_n$. A projection cone is the cone of data vectors that orthogonally project onto a given non-degenerate equidistant tree.

Projection-cone bound conjecture. Every maximal cell in $\mathcal{Q}_n$ is obtained as the intersection of at most $(n-1)!$ projection cones.

The preceding theorem shows that some maximal cells are intersections of at least $(n-1)!$ projection cones, so the conjecture would make this bound sharp. The source gives no proof of the upper bound.