The three-species reversible–irreversible network conjecture

Let $G$ be a reaction network consisting of one reversible-reaction pair $y\leftrightarrows y'$ and one irreversible reaction $\widetilde y\to\widetilde y'$. The reaction vectors are $y'-y$ and $\widetilde y'-\widetilde y$. For a species $i$, let the embedded network of $G$ obtained by removing all species except $i$ be the resulting one-species network; call it a 2-alternating network when it has the form $\leftrightarrows\,\,\to$ or $\leftarrow\,\,\leftrightarrows$. Three-species reversible–irreversible network conjecture. The network $G$ is nondegenerately multistationary if and only if the reaction vectors are nontrivial scalar multiples of each other, namely

$$y'-y=\lambda(\widetilde y'-\widetilde y)$$

for some $0\neq\lambda\in\mathbb{R}$, and, for some species $i$, the embedded network is a 2-alternating network. The conjecture is presented as a future goal for extending the paper's results beyond two species; the source gives no evidence that it has been resolved.