Vojta's conjecture for smooth projective schemes

Let $k$ be a number field. Let $X$ be a smooth projective scheme over $k$, let $H$ be a big line bundle on $X$, let $r$ be a positive integer, and fix $\delta>0$. For a closed point $x\in X$, write $k(x)$ for its residue field, $h_{K_X}(x)$ for the height associated with the canonical bundle, $h_H(x)$ for the height associated with $H$, and $d_k(k(x))$ for the relative logarithmic discriminant. Then there exists a proper Zariski closed subset $Z\subset X$ such that, for all closed points $x\in X$ with $x\notin Z$ and $[k(x):k]\leq r$,

$$h_{K_X}(x)-\delta h_H(x)\leq d_k(k(x))+O(1).$$

Vojta's conjecture. The stated inequality holds for all such closed points outside $Z$.

This is a height-theoretic conjecture extending Vojta's framework to the setting needed for the paper, and the paper notes that it implies a version for algebraic stacks. Its status is not resolved in the supplied text.