Product-form Poisson conjecture for binary reaction networks

Let a reaction network have complexes $y$ satisfying $\\|y\\\|_1\le 2$, and let the initial distribution of its stochastically modeled process be a product of Poisson distributions. Let $u(t)$ solve the mass-action equation, and let $B(u)$ be the diffusion matrix defined by

$$B_{ij}(u)=\sum_k\kappa_k u^{y_k}\bigl(y'_{ki}y'_{kj}-y_{ki}y_{kj}-\delta_{i,j}\zeta_{ki}\bigr).$$

Product-form Poisson conjecture. If $B(u(t))=0$, then the distribution of the process $X_t$ is a product of Poisson distributions for all time.

This conjecture asserts that, for binary reaction networks with Poisson product-form initial data, vanishing diffusion in the associated Fokker–Planck representation preserves the product-form Poisson law along the deterministic mass-action trajectory. The source presents this as a conjectural consequence of the Poisson representation; its resolution is not established in the supplied text.