The quadratic-zero conjecture for positive semidefinite quartic forms

Let $P_{n,4}$ denote the space of homogeneous quartic forms on $\mathbb{R}^n$, and let $G\in P_{n,4}$ be positive semidefinite if $G(x)\geq 0$ for all $x\in\mathbb{R}^n$. Suppose the set of spherical zeros of $G$ is a quadratic variety and a connected full submanifold of $\mathbb{R}^n$ with positive dimension and positive codimension in the sphere. The quadratic-zero conjecture. Then $G$ is a sum of squares of quadratic forms. This conjecture would provide a criterion ensuring that a positive semidefinite quartic form is sos from the geometry of its spherical zero set; the supplied status evidence gives a counterexample for positive semidefinite degree-$6$ forms with cubic spherical zeros, not for the quartic claim.