Bogomolov conjecture for powers of the multiplicative group

Let $V\subset {\mathbb G}_m^N$ be an irreducible subvariety defined over $\overline{\mathbb Q}$. For each $\epsilon\geq 0$, set

$$S_{\epsilon}:=\left\{P\in\left(\overline{\mathbb Q}^{*}\right)^N: h(P)\leq\epsilon\right\}.$$

Bogomolov conjecture. If, for each $\epsilon>0$, the set $V(\overline{\mathbb Q})\cap S_{\epsilon}$ is Zariski dense in $V$, then $V$ is a torsion translate of an algebraic subgroup of ${\mathbb G}_m^N$.

This strengthens the torsion-point case of Laurent's theorem by replacing height zero with arbitrarily small height. The conjecture for every power of the multiplicative group was proved by Zhang.