The extendability and weak SOS-hyperbolicity conjecture

Let $\Gamma_{n,d}$ be the class of symmetric degree-$d$ polynomials in $n$ variables. For a symmetric hyperbolic polynomial $p$, let its associated operator be the operator defined in the paper; call this operator extendable when it has the stated extension property, and call $p$ weakly SOS-hyperbolic when it satisfies the paper's weak sum-of-squares hyperbolicity condition. Extendability conjecture. If $p\in\Gamma_{n,d}$ is symmetric hyperbolic, then the associated operator of $p$ is extendable if and only if $p$ is weakly SOS-hyperbolic. This is presented as a more speculative conjecture, and no general proof or disproof is given.