Rémond's conjecture on heights in divisible hull extensions

Let $\mathbb{K}$ be a number field, let $\Gamma\leq\mathbb{K}^{\times}$ be a non-torsion finitely generated subgroup, and define its divisible hull by

$$\Gamma^{\mathrm{div}}=\{\alpha\in\overline{\mathbb{Q}}^{\times}:\alpha^n\in\Gamma\text{ for some }n\in\mathbb{N}\}.$$

Put $\mathbb{L}=\mathbb{K}(\Gamma^{\mathrm{div}})$. Rémond's conjecture. The Weil height is bounded from below on $\mathbb{L}^{\times}\setminus\Gamma^{\mathrm{div}}$. The conjecture would describe all small-height elements in this extension as arising from the divisible hull; the paper states that it lies beyond the scope of its methods and remains open.