The facet characterization conjecture for the cone of totally balanced games

Let $N$ be a finite set, and let $E(N)$ denote the cone of totally balanced games on $N$. A min-balanced system is a set system whose associated inequality is min-balanced; it is irreducible if it has no proper subsystem defining the same type of inequality, and non-trivial if it is not a trivial system. For a min-balanced system ${\cal B}$ with $\bigcup {\cal B}\subset N$, let its associated inequality be the corresponding min-balanced inequality, and call the inequality obtained by the conjugation operation the conjugate inequality.

Facet characterization conjecture. If $|N|\geq 3$, then the facet-defining inequalities for the cone $E(N)$ are exactly the inequalities corresponding to non-trivial irreducible min-balanced systems ${\cal B}$ with $\bigcup {\cal B}\subset N$, together with their conjugate inequalities.

This conjecture gives a complete description of the facets of the cone of totally balanced games. The authors state that it is known to be true for $|N|\leq 5$; its validity for general finite sets $N$ remains open.