Persistence Conjecture for arithmetic hyperbolicity of algebraic stacks

Let $k$ be an algebraically closed field of characteristic zero, and let $X$ be a finitely presented algebraic stack over $k$. Recall that $X$ is arithmetically hyperbolic over $k$ if there is a ${\mathbb Z}$-finitely generated subring $A\subseteq k$ and a finitely presented model $\mathcal{X}$ for $X$ over $A$ such that, for every ${\mathbb Z}$-finitely generated subring $A'\subseteq k$ containing $A$, the set

$$\operatorname{im}\left(\pi_0(\mathcal{X}(A'))\to\pi_0(\mathcal{X}(k))\right)$$

is finite. It is absolutely arithmetically hyperbolic if $X_L$ is arithmetically hyperbolic over $L$ for every algebraically closed field extension $L\supseteq k$.

Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is absolutely arithmetically hyperbolic.

This conjecture asserts that arithmetic hyperbolicity persists after every algebraically closed field extension. It is formulated for finitely presented algebraic stacks and is relevant to the passage from arithmetic hyperbolicity over $\overline{\mathbb Q}$ to absolute arithmetic hyperbolicity.