Vojta's conjecture with multiplier ideal sheaves

Let $\mathcal X$ be a smooth projective variety defined over $\mathbb Q$, with canonical divisor $K_\mathcal X$, ample divisor $\mathcal A$, and nonzero ideal sheaf $\mathcal I$. Let $S$ be a finite set of places of $\mathbb Q$ containing $M_\mathbb Q^\infty$. Write $\mathcal I^-$ for the associated limiting multiplier ideal sheaf, and let $h_{K_\mathcal X}$, $h_\mathcal A$, $\lambda_\mathcal I$, and $\lambda_{\mathcal I^-}$ denote the corresponding height functions.

Vojta's multiplier-ideal conjecture. For every real $\varepsilon>0$ and positive integer $r$, there is a proper Zariski-closed subset $Z\subsetneq\mathcal X$, depending only on $\mathcal X,\mathcal I,\mathcal A,\varepsilon,r$, such that

$$h_{K_\mathcal X}(\mathbf x)+\sum_{\nu\in S}\lambda_\mathcal I(\mathbf x,\nu)-\sum_{\nu\in S}\lambda_{\mathcal I^-}(\mathbf x,\nu)\leq\varepsilon h_\mathcal A(\mathbf x)+d(\mathbf x)+O(1),$$

for all $\mathbf x\in(\mathcal X\setminus Z)(\overline{\mathbb Q})$ with $[\mathbb Q(\mathbf x):\mathbb Q]\leq r$.

This is a multiplier-ideal formulation of Vojta's conjecture, replacing normal-crossings data by a correction term. Its general status is open.