The two-dimensional integer-hull conjecture without the linear-growth assumption

Let $d=2$, and consider the setting of Theorem~, concerning integer hulls of linear polyhedra in dimension $d$.

Two-dimensional integer-hull conjecture. In dimension $d=2$, Theorem~ holds without the $O(n)$ assumption.

The question is motivated by the difficulty of extending the theorem beyond the $O(n)$ setting: in dimension two, the relationship between integer hulls and continued fractions suggests that the theorem should remain valid, while the corresponding assertion in dimensions three and higher is unresolved and has no known counterexample.