In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell conjecture, and in higher dimensions, the Lang conjecture states that if $X(\mathbb{C})^{an}$ is hyperbolic, then $|X(k)|$ is finite. I would be interested to generalize this train of thought to the category of schemes up to $\mathbb{A}^1$-homotopy. Concretely, my question is whether or not there exists a (conjectural) geometrical condition on $[X]$, an $\mathbb{A}^1$-homotopy class, such that the set of morphism $\text{Hom}_{\mathbb{A}^1\text{-hom}}([\text{Spec } k],[X])$ in the category of schemes up to $\mathbb{A}^1$-homotopy is finite.
A naive translation of Faltings Theorem of course doesn't work, as for instance $\mathbb{A}^1_{\mathbb{Q}}$ has infintely many rational points, but $\text{Spec }\mathbb{Q}$ does not.
