Kodaira-dimension conjecture for dense p-adic analytic maps

Let $X$ be a smooth projective variety over $\mathbb{C}_p$, and let $X^{\mathrm{an}}$ denote its analytification. Kodaira-dimension conjecture. If there is a $p$-adic analytic map

$$f:\mathbb{C}_p\to X^{\mathrm{an}}$$

with Zariski-dense image, then the Kodaira dimension satisfies $\kappa(X)=-\infty$. This strengthens the stated conjecture of Cherry for $K3$ surfaces and predicts that a dense $p$-adic analytic map can occur only for varieties of negative Kodaira dimension. Its status is open.