The Diophantine approximation conjecture for global fields

Let $K$ be a global field and let $v$ be a place of $K$. Let $X$ and $Y$ be positive dimensional irreducible projective varieties over $K$, let $f:X\dashrightarrow Y$ be a dominant rational map defined over $K$, and let $U$ be a non-empty Zariski open subset of $X$ defined over $K$ and contained in the domain of $f$. Suppose that $X(K)$ is Zariski dense in $X$. Diophantine approximation conjecture. There is an effective Cartier divisor $D$ on $Y$ defined over $K$ such that $\lambda_{Y,D,v}$ is unbounded on $f(U(K))-\operatorname{supp}(D)$. Equivalently, there is a sequence of $K$-rational points in $U-f^{-1}(\operatorname{supp}(D))$ whose images under $f$ approach $\operatorname{supp}(D)(K_v)$ $v$-adically. This proposes a general approximation principle for rational points on global varieties and local Weil functions.