Nondegeneracy conjecture for positive steady states in reaction networks

A reaction network has positive steady states when, for suitable rate constants and a stoichiometric compatibility class, its mass-action system has steady states in which every concentration is strictly positive. Let $\cap_{pos}(G)$ be the maximum number of positive steady states admitted by a network $G$, and let $\cap_{nondeg}(G)$ be the maximum number of those positive steady states that are nondegenerate, meaning that the image of the Jacobian restricted to the stoichiometric subspace equals that subspace.

Nondegeneracy conjecture. Given a network $G$, if

$$\cap_{pos}(G)<+\finite,$$

then

$$\cap_{nondeg}(G)= bsp \cap_{pos}(G).$$

The conjecture asserts that whenever the number of positive steady states is finite, all of them can be realized as nondegenerate steady states, in the sense of the corresponding maximum capacities. It is proved for small networks with one species and for a sub-family of networks with at most two reactions, possibly reversible; its general status is not resolved in the source.