Joshi–Shiu's multistationarity conjecture for one-dimensional reaction networks

Let a reaction network have a one-dimensional stoichiometric subspace, and call a complex a source complex if it is the source of at least one reaction. A one-species embedded subnetwork is obtained by restricting the network to one species and retaining the induced reaction-pattern information; the patterns $(\to,\gets)$ and $(\gets,\to)$ indicate the corresponding directions of reactions in that subnetwork.

Joshi–Shiu's conjecture. A reaction network with one-dimensional stoichiometric subspace and more than one source complex has the capacity for multistationarity if and only if it has a one-species embedded subnetwork with the pattern $(\to,\gets)$, and another (possibly the same) with pattern $(\gets,\to)$.

The paper states that its results answer this conjecture in the affirmative, so the criterion is established for the class of networks under consideration.