Product-form characterization for conservative weakly reversible reaction networks

Let $(\mathcal{G},\kappa)$ be a conservative weakly reversible chemical reaction network under mass-action kinetics. Write $\mathcal{R}$ for its set of reactions, and for each reaction $r\in\mathcal{R}$ let $\nu_r$ and $\nu'_r$ denote its reactant and product complexes. Assume that for every $r\in\mathcal{R}$ there is no distinct reaction $\tilde r\in\mathcal{R}$ with

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

Product-form characterization conjecture. The network $(\mathcal{G},\kappa)$ has a product-form stationary distribution if and only if it is complex balanced.

This conjecture proposes that, under the stated nondegeneracy condition on reaction differences, product-form stationary distributions occur only through complex balance, although the paper exhibits weakly reversible positive-deficiency networks with rate-independent product forms beyond complex balance. Its status is not resolved in the supplied text.