S4-Conjecture for ternary quartics and quadratic forms

Let $f$ be a ternary quartic, that is, a $4$-form in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2,\mathrm{x}_3]$, and let $g$ be a quadratic form in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2,\mathrm{x}_3]$. Suppose there exists $x'\in\mathbb{R}^3$ such that $g(x')>0$. Write

$$S(h)=\{x\in\mathbb{R}^3:h(x)\geq 0\}$$

for the non-negativity set of a polynomial $h$. S4-Conjecture. The following statements are equivalent: (a) $S(g)\subseteq S(f)$; (b) there exists a non-negative homogeneous polynomial $t\in\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2,\mathrm{x}_3]_2$ such that

$$f(x)-t(x)g(x)\geq 0$$

for all $x\in\mathbb{R}^3$. The conjecture is resolved: the source states that this question was answered by the author. This result provides a higher-degree analogue of the homogeneous S-lemma, replacing a non-negative scalar multiplier by a non-negative homogeneous quadratic polynomial.