Cherry's hyperbolicity equivalence conjecture for non-Archimedean analytic spaces

Let $k$ be a non-Archimedean complete valued field that is nontrivially valued and algebraically closed, and let $X$ be a smooth compact boundaryless $k$-analytic space. An entire curve in $X$ is an analytic map from $\mathbb{A}^{1,\mathrm{an}}$ to $X$, and $\mathrm{Mor}_k(\mathbb{D},X)$ denotes the family of analytic maps from the unit disk $\mathbb{D}$ to $X$. The space $X$ is Cherry-Kobayashi hyperbolic when its Cherry-Kobayashi semidistance is an actual distance. Cherry's hyperbolicity equivalence conjecture. The following conditions are equivalent: (1) $X$ is Cherry-Kobayashi hyperbolic; (2) $X$ contains no entire curve; (3) $X$ contains no rational curve; (4) the family $\mathrm{Mor}_k(\mathbb{D},X)$ is normal. This conjecture proposes a non-Archimedean analogue of equivalences between hyperbolicity, the absence of entire or rational curves, and normality. Cherry established related results for abelian varieties and projective curves, but the equivalence in this generality remains open.