The conjecture on intersections of subvarieties with tori

Let $n\in \mathbb{N}$, let $W\subseteq \mathbb{G}_{\mathrm{m}}^n$ be an irreducible subvariety, and let $W^{\mathrm{atyp}}$ denote the union of the atypical components of $W\cap H$ over all algebraic subgroups $H\subseteq \mathbb{G}_{\mathrm{m}}^n$. A component $X$ of $W\cap H$ is atypical when

$$\dim X>\dim W+\dim H-n.$$

The conjecture on intersections of subvarieties with tori. For every $n\in\mathbb{N}$ and every irreducible subvariety $W\subseteq\mathbb{G}_{\mathrm{m}}^n$, the set $W^{\mathrm{atyp}}$ is a proper Zariski-closed subset of $W$.

This is a diophantine conjecture about atypical, also called anomalous or unlikely, intersections of varieties with algebraic subgroups of tori. The supplied source does not state whether the conjecture has been resolved.