Tree-model multiplicity conjecture

Let $\mathcal{M}=(G,In,Out,Leak)$ be a linear compartment model with input, output, and leak in compartment $1$ only, so that $In=Out=Leak=\{1\}$, and suppose that $G$ is a bidirectional tree. Apply the stated procedure: label outgoing leaf-edges by $0$ and recursively label each outgoing edge by one plus the sum of the labels on its outgoing successor edges; label incoming leaf-edges by $1$ and recursively label each incoming edge by one plus the sum of the labels on all edges incoming to its tail compartment. This assigns an integer to each edge $(i,j)$, intended as the multiplicity of $a_{ji}$. Tree conjecture. The procedure above yields the multiplicities of the parameter variables $a_{ji}$ in the equation of the singular locus. The conjecture is attributed to Molly Hoch, Mark Sweeney, and Hwai-Ray Tung in personal communication. No resolution is supplied in the paper.