Rationality conjecture for primitive algebraic compactifications of $\mathbb{C}^2$

Let $\bar X$ be a primitive algebraic compactification of $\mathbb{C}^2$ formed by minimally resolving the singularities of a curve-germ at a point on the line $L_\infty$ at infinity on $\mathbb{P}^2$, then contracting the strict transform of $L_\infty$ and all exceptional curves other than the last one. Let $P_\infty$ be the point at infinity from the algebraic-place construction, and let $g$ be the smallest integer such that there exists a curve $C$ on $\bar X$ with geometric genus $g$ that does not pass through $P_\infty$. Rationality conjecture. In this situation, $g=0$ if and only if the singularity of $\bar X$ at $P_\infty$ is rational. The statement is a conjectural answer to the first case of the question relating the smallest genus of a curve avoiding $P_\infty$ to the singularity at $P_\infty$; the source reports computed examples supporting it, but gives no resolution.