Dehomogenized S4-Conjecture for bivariate degree-four and degree-two polynomials

Let $f$ be a polynomial of degree $4$ in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2]$ and let $g$ be a polynomial of degree $2$ in $\mathbb{R}[\mathrm{x}_1,\mathrm{x}_2]$. Suppose there exists $x'\in\mathbb{R}^2$ such that $g(x')>0$. Write

$$S(h)=\{x\in\mathbb{R}^2:h(x)\geq 0\}$$

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

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

for all $x\in\mathbb{R}^2$. The conjecture is refuted: the source gives explicit counterexamples, including $\tilde f=\mathrm{x}_1^3+\mathrm{x}_1^3\mathrm{x}_2+\mathrm{x}_2^2$ and $\tilde g=\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_1\mathrm{x}_2$.