Jordan-algebra representation conjecture for convex state spaces

Let $C$ be a finite-dimensional convex compact set. A local regret function is a regret function defined locally on $C$, and a transitive symmetry group is a group of symmetries acting transitively on $C$.

Jordan-algebra representation conjecture. If $C$ has a local regret function and a transitive symmetry group, then $C$ can be represented as the positive elements of a Jordan algebra with trace $1$.

This conjecture connects local Bregman-type information geometry and transitive symmetry with the Jordan-algebraic description of state spaces. The source presents it as a conjecture but gives no resolution or further conditions beyond finite dimensionality, compactness, locality of the regret function, and transitivity.