Koponen's conjecture on primitive binary finitely homogeneous supersimple structures

A structure is primitive if no nontrivial equivalence relation on its domain is invariant under its automorphism group; it is binary if its language has relation symbols of arity at most two, finitely homogeneous if every isomorphism between finite substructures extends to an automorphism, and supersimple if its first-order theory is supersimple. A random structure is a structure obtained by the generic construction for a class of finite structures with the appropriate amalgamation properties. Koponen's conjecture. Any primitive binary finitely homogeneous supersimple structure is a random structure. The conjecture proposes that, within this class, the random graph provides the canonical model up to the relevant notion of equivalence; the source gives no resolution, so its status remains open.