Might the following be true:

Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the following are equivalent:

1) $q(x,y,z)\geq 0\Rightarrow p(x,y,z)\geq 0$ for all $x,y,z\in \mathbb{R}$

2) there are nonnegative homogeneous polynomials $s,t$ with $\deg(s)\leq 4$ and $\deg(t)\leq 2$ such that $p=s+qt$.

Background:

A) The case of the above conjecture that $\deg(p)=2$ is exactly the S-lemma for polynomials $p,q$ in three variables.

B) The analogous statement for homogeneous polynomials $p,q$ in *two* variables is also known to be true.

C) By a theorem of Hilbert, a homogeneous polynomial $p\in\mathbb{R}[x,y,z]$ of degree 4 is nonnegative if and only if it is a sum of three squares of quadratic polynomials.

If the above conjecture holds up then $$\min\{p(x,y)\mid q(x,y)\geq 0\}$$ for inhomogeneous polynomials $p,q$ with $\deg(p)=4$ and $\deg(q)=2$ can be cast as a semidefinite optimization problem.