Sum-of-squares conjecture for isotropic RR discriminant factors

Let $V$ be the variety under consideration, let its RR discriminant be defined by a polynomial, and call a factor isotropic when it belongs to the isotropic part of that discriminant. Sum-of-squares conjecture. Every irreducible isotropic factor of the defining polynomial of the RR discriminant is a sum of squares. The conjecture has been verified in the examples of the rational quadric, the twisted cubic, and projective space; it remains open in general.