Duality conjecture for preclusters and maximal invariant polyhedral supports

A mass-action network is a chemical reaction network whose dynamics is of mass-action type. A network is conservative when it has a nontrivial conservation law, and a positive steady state is a steady state with all species concentrations positive. A precluster is a connected component of the preclustering graph, and a maximal invariant polyhedral support is a maximal polyhedral support invariant under the network dynamics.

Precluster–support duality conjecture. For conservative mass-action networks that do not have two species with exactly the same rates and have at least one positive steady state, there is a duality relation between the sets of preclusters and of maximal invariant polyhedral supports.

This conjecture proposes a combinatorial duality between preclusters and invariant polyhedral supports, extending the duality between conserved quantities and internal cycles established for conservative mass-action networks. The source does not state whether the conjecture has been resolved.