Geometric Bombieri--Lang's special-locus conjecture

Let $K/k$ be a finitely generated extension of fields of characteristic zero, with $k$ algebraically closed in $K$. Define the algebraic special set $\operatorname{Sp}(X)$ as the Zariski closure of the images of all non-constant rational maps from abelian varieties over $\overline K$ to $X_{\overline K}$. For a projective variety $Z$ over $K$, a constant model is a birational $K$-map $T_K\dashrightarrow Z$ from a projective variety $T$ over $k$, and its induced points are obtained on its maximal morphism locus from $T(k)$. Geometric Bombieri--Lang's special-locus conjecture. If $Z$ is the Zariski closure in $X$ of $(X\setminus\operatorname{Sp}(X))(K)$, then there are finitely many distinct closed subvarieties $Z_1,\ldots,Z_r$ of $Z$, containing all irreducible components of $Z$, such that each $Z_i$ is birationally constant and every point of $(X\setminus\operatorname{Sp}(X))(K)$ is induced by one of these constant models. This is the more precise function-field formulation and remains open in general.