Positive-integral-point equidistribution conjecture for real genus-zero curves

Let $C\subset\mathbb{C}^2$ be a curve defined over $\mathbb{Z}$ and irreducible over $\mathbb{C}$. Let $C_{\mathbb{R}}$ be an irreducible component of $C\cap\mathbb{R}^2$ whose intersection with the first quadrant is noncompact. Positive-integral-point conjecture.

$$C_{\mathbb{R}}\text{ has infinitely many integral points}\Longrightarrow C_{\mathbb{R}}\text{ has infinitely many positive integral points}.$$

The conjecture is motivated by the desired equivalence between computability of \mathbf{\Big}_{\mathbb{N},2} and \mathbf{\Big}_{\mathbb{Z},2}; together with quantifier elimination over $\mathbb{R}$, it would imply that equivalence. Its status is open in the source.