The diameter-c0 radial-or-one-dimensional conjecture for the Liouville equation

Let $u$ be a solution of the Liouville equation

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

in $\mathbb{R}^2$, and let $\delta$ be the standard Euclidean metric. Assume that $\mathbb{R}^2$ has diameter $pi$ under the metric $e^{2u}\delta$. The diameter-c0 conjecture. Then $u$ must be either radial about a point or one-dimensional. This is one of the two rigidity questions that the authors state remain open in full generality; the conjecture seeks a classification under a geometric diameter condition.