The unlikely intersection bound with multiplicities

Let $K$ be a number field, let $\eta$ be the generic point of $\operatorname{Spec}\mathcal{O}_K$, and let $\mathcal{V}\subseteq\mathbb{G}^n_{m,\mathcal{O}_K}$ be an integral closed subscheme dominating $\operatorname{Spec}\mathcal{O}_K$. Put $V=\mathcal{V}_\eta$. Suppose that $V$ is not contained in any proper algebraic subgroup of $\mathbb{G}^n_{m,K}$. For a flat subgroup scheme $\mathcal{H}$, write $\mathcal{C}(\mathcal{H})$ for its complexity, and let $\operatorname{mult}(\mathcal{Z})$ denote the multiplicity of an irreducible component $\mathcal{Z}$ of the intersection.

Unlikely intersection bound. For every $\varepsilon>0$, there exists $c=c(n,\mathcal{V},\varepsilon,K)\in\mathbb{R}$ such that, for every flat subgroup scheme $\mathcal{H}\subseteq\mathbb{G}^n_{m,\mathcal{O}_K}$ satisfying $\dim\mathcal{H}+\dim\mathcal{V}<n+1$,

$$\sum_{\mathcal{Z}}\operatorname{mult}(\mathcal{Z})\deg(\mathcal{Z}_{\operatorname{red}})\log N(\mathcal{Z})\leq\varepsilon\mathcal{C}(\mathcal{H})^{\dim\mathcal{V}}+c.$$

The sum is over all irreducible components of $\mathcal{V}\cap\mathcal{H}$ that do not dominate $\operatorname{Spec}\mathcal{O}_K$. This is the unlikely-intersection refinement of the general bound: the generic-fiber noncontainment hypothesis and strict dimension inequality permit an arbitrarily small coefficient of the complexity term. The source presents it as conjectural and discusses particular instances and related known results.