The mild boundedness conjecture for arithmetically hyperbolic varieties

Throughout, $k$ is an algebraically closed field of characteristic zero, and a variety over $k$ is a finite type separated integral $k$-scheme. A variety $X$ over $k$ is arithmetically hyperbolic if, for every $\mathbb{Z}$-finitely generated subring $A\subset k$ and every finite type separated model $\mathcal{X}$ over $A$ with $\mathcal{X}_k\cong X$, the set $\mathcal{X}(A)$ is finite. Mild boundedness conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is mildly bounded over $k$. This proposed strengthening would imply that arithmetic hyperbolicity persists after arbitrary extensions of the base field. The source presents it as an interesting conjecture and gives no resolution.