The affine-surface higher-cancellation conjecture

Let $X$ be a smooth affine surface. Say that $X$ has negative logarithmic Kodaira dimension when its logarithmic Kodaira dimension is negative. Say that $X$ fails the $\mathbb{A}^{n}$-cancellation property if there exists a smooth affine surface $Y$ with $X\times\mathbb{A}^{n}\simeq Y\times\mathbb{A}^{n}$ but $X\not\simeq Y$. Higher-cancellation conjecture. A smooth affine surface $X$ with negative logarithmic Kodaira dimension is either isomorphic to the total space of a line bundle over a curve, or it fails the $\mathbb{A}^{2}$-cancellation property. Furthermore, every non-rigid $X$ which fails the $\mathbb{A}^{2}$-cancellation property also fails the $\mathbb{A}^{1}$-cancellation property. The proposed strengthening is intended to characterize stabilization behavior of smooth affine surfaces and would settle the question of their behavior under stabilization by affine spaces; its status is not resolved in the supplied source.