Dividing-edge conjecture for linear compartmental models

Let $\mathcal{M}$ be a strongly connected linear compartmental model that is generically locally identifiable, with singular-locus equation $f$. An edge $j\to i$ is dividing if its parameter $k_{ij}$ divides $f$. Dividing-edge conjecture. If $j\to i$ is a dividing edge of $\mathcal M$, and the model $\mathcal{M}'$ obtained by deleting the edge $j\to i$ is strongly connected, then $\mathcal{M}'$ is unidentifiable. The conjecture is motivated by the converse of the known result that deleting non-dividing edges preserves generic local identifiability. It is proved in cases where deleting the edge increases the shortest input-output path length, but the general claim remains open.