Geometric Bombieri--Lang's generic-sequence height conjecture

Let $K=k(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety of general type over $K$. A sequence of $K$-rational points in $X$ is generic if every infinite subsequence is Zariski dense in $X$. Geometric Bombieri--Lang's generic-sequence conjecture. Every generic sequence of $K$-rational points in $X$ has bounded height. This is presented as an equivalent height formulation and remains open in general.