The global attractor conjecture for weakly reversible reaction networks

Let a reaction network be weakly reversible, let its rate constants be positive, and consider its rate equation. A positive stoichiometric compatibility class consists of the positive vectors in a stoichiometric compatibility class.

Global attractor conjecture. The rate equation has exactly one equilibrium solution in each positive stoichiometric compatibility class, and any positive solution that starts in the same class will approach this equilibrium as $t\to +\infty$.

This conjecture asks whether the local convergence supplied by the zero deficiency theorem extends to every positive initial condition in the same class. It is a famous open question; the conjecture was first raised by Horn in 1974 and was proved only in special cases in the cited work.