The Persistence Conjecture for arithmetically hyperbolic schemes

Let $k$ be an algebraically closed field of characteristic zero. A finite type separated scheme $X$ over $k$ is arithmetically hyperbolic if, for every $\mathbb{Z}$-finitely generated subring $A\subset k$ and every finite type separated scheme $\mathcal{X}$ over $A$ with $\mathcal{X}_k\cong X$, the set $\mathcal{X}(A)$ is finite. Let $L/k$ be an extension of algebraically closed fields of characteristic zero. Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X_L$ is arithmetically hyperbolic over $L$. This predicts that arithmetic hyperbolicity persists under extensions of algebraically closed characteristic-zero fields; it is known in several cases, including varieties with suitable maps to semi-abelian varieties, but remains open in general.