Non-Archimedean uniruledness conjecture for analytifications

Let $K$ be a complete discrete valued field with equal characteristic $0$, and let $\mathcal{X}$ be a uniruled variety over $K$. Say that a proper geometrically integral smooth $K$-analytic space is non-Archimedean uniruled if it can be covered by open virtual disks after a finite field extension $K'|K$. Non-Archimedean uniruledness conjecture. The analytification $\mathcal{X}^{\textsf{an}}$ is non-Archimedean uniruled; equivalently, it can be covered by open virtual disks after a finite field extension $K'|K$. This is proposed as the non-Archimedean analogue of uniruledness, motivated by the expectation from Mori's conjecture and by the emptiness of the essential skeleton for smooth uniruled varieties. It remains open in general.