The SOS characterization of symmetric hyperbolicity

Let $\Gamma_{n,d}$ denote the class of symmetric degree-$d$ polynomials in $n$ variables, and let $\Delta_{\vec{1},\vec{1}}p$ be the indicated second directional derivative. A polynomial is SOS if it is a sum of squares of polynomials. SOS characterization conjecture. For every $p\in\Gamma_{n,d}$, $p$ is symmetric hyperbolic if and only if $\Delta_{\vec{1},\vec{1}}p$ is SOS. The conjecture is motivated by computational evidence and by the relationship between the SOS condition and symmetric hyperbolicity; its general validity is open.