Spectral transitive-symmetry conjecture for simple Jordan algebras

Let $S$ be a finite-dimensional convex compact set that is spectral, meaning that all of its states have orthogonal decompositions with a common spectrum, and suppose that the symmetry group of $S$ acts transitively. Spectral transitive-symmetry conjecture. $S$ can be represented as the set of positive elements of a simple Jordan algebra having trace $1$. This conjecture proposes that spectrality together with transitive symmetry characterizes normalized positive elements of simple Jordan algebras; the source gives no resolution or further evidence for the claim.