Let $f(x, y)$ be a homogeneous polynomial with real coefficients in $2$ determinates $x , y$. Suppose that $$f(x, y) \equiv g(x, y)^2 \pmod{x^2 + y^2 - 1}$$ for some polynomial $g(x, y)$, where $g(x, y)$ is not necessarily homogeneous.
Is it true that there exists homogeneous polynomials $h_1(x, y), \dots, h_n(x, y)$ such that $$f(x, y) \equiv h_1(x, y)^2 + \cdots + h_n(x, y)^2 \pmod{x^2 + y^2 - 1}? \tag{*}$$
Partial Motivation. In real algebraic geometry, there is a theorem of Schmuedgen that applies for any compact semialgebraic set $X \subset \mathbb{R}^N$. In the case where $X = S^1$ is the unit sphere in $\mathbb{R}^2$, this theorem states that every polynomial $f(x, y) \in \mathbb{R}[x, y]$ that is strictly positive on $S^1$ can be written as a sum of squares of functions on $S^1$, i.e. $$f(x, y) \equiv h_1(x, y)^2 + \cdots + h_n(x, y)^2 \pmod{x^2 + y^2 - 1} \text{ for some } h_1, \dots, h_n \in \mathbb{R}[x, y]. \tag{o}$$
On the other hand, if such $f(x, y)$ that is strictly positive on $S^1$ is additionally assumed to be homogeneous, then by a weak version of another theorem of Reznick, these $h_1, \dots, h_n$ can be chosen to be homogeneous. In particular, Reznick's Theorem implies Schmuedgen's theorem for homogeneous $f$.
Now suppose that the answer to my question is "yes". Then this affirmative answer implies that every homogeneous polynomial representable as in (*) with $n = 1$ can be represented as in (o). In this sense, my question asks for a partial converse that Schmuedgen's Theorem implies Reznick's Theorem.