The concave Liouville solution one-dimensionality conjecture

Let $u$ be a solution of the Liouville equation

$$-\Delta u=e^{2u}$$

in $\mathbb{R}^2$. The concave-solution conjecture. If $u$ is concave, then $u$ is one-dimensional. The paper proves the analogous flat-level-set rigidity result in higher dimensions but states that the corresponding two-dimensional questions, including this one, remain open in full generality.