Leak-addition conjecture for unidentifiable linear compartmental models

Let $\mathcal{M}$ be an unidentifiable linear compartmental model. Leak-addition conjecture. If one leak is added, the resulting model $\widetilde{\mathcal{M}}$ is also unidentifiable. This is equivalent to the assertion that removing a leak from an identifiable model preserves identifiability, in the corresponding setting. The paper proves special cases, but the general conjecture remains open.