Real-point conjecture for separating extreme rays of nonnegative ternary and quaternary quartics

Let $P_{3,6}$ and $P_{4,4}$ denote the cones of nonnegative homogeneous polynomials of the indicated degrees in three and four variables, respectively, and let $\Sigma_{3,6}$ and $\Sigma_{4,4}$ denote the corresponding sums-of-squares cones. For a polynomial outside the relevant sums-of-squares cone, a separating extreme ray is understood in the sense of the cited theorem.

Real-point conjecture. For $p \in P_{3,6} \setminus \Sigma_{3,6}$ there exist $v_1,\ldots,v_9 \in \mathbb{R}^3$ yielding a separating extreme ray for $p$. Analogously, for $p \in P_{4,4} \setminus \Sigma_{4,4}$, there exist $v_1,\ldots,v_8 \in \mathbb{R}^4$ yielding a separating extreme ray for $p$.

This conjecture strengthens Blekherman's conjecture by requiring the points describing the relevant extreme rays to be real. The paper presents it as a formulation based on its results; its resolution is not specified in the supplied text.