Non-product-form conjecture for conservative non-weakly reversible reaction networks

Let $\mathcal{G}$ be a conservative, almost essential, non-weakly reversible chemical reaction network under mass-action kinetics. Let $\mathcal{R}$ be its set of reactions, and let $\mathcal{R}'\subseteq\mathcal{R}$ be a connected component that is not weakly reversible. For each reaction $r$, write $\nu_r$ and $\nu'_r$ for its reactant and product complexes. Assume that there is a reaction $r\in\mathcal{R}'$ such that no reaction $\tilde r\in\mathcal{R}\setminus\mathcal{R}'$, with $\tilde r\ne r$, lies in a weakly reversible component and satisfies

$$\nu'_r-\nu_r=\nu'_{\tilde r}-\nu_{\tilde r}.$$

Non-product-form conjecture. Then $\mathcal{G}$ has a non-product-form stationary distribution independently of the reaction rate; equivalently, it belongs to class N.

The conjecture is motivated by the non-weakly reversible examples in the paper, for which product-form stationarity fails independently of the rate. No resolution is given in the supplied text.