The converse conjecture for deleting one leak in linear compartmental models

Let $\widetilde{\mathcal M}$ be a linear compartmental model that is strongly connected and has at least one input and exactly one leak. Let $\mathcal M$ be the model obtained from $\widetilde{\mathcal M}$ by removing the leak. Deleting-one-leak conjecture. If $\widetilde{\mathcal M}$ is generically locally identifiable from the coefficient map, then so is $\mathcal M$. This conjecture is the converse of the theorem establishing that adding one leak preserves generic local identifiability; its resolution would characterize identifiability under deletion of a single leak in this class of models.