  Let $\Gamma\subset \bar{\mathbb{Q}}^*$ be a finitely generated subgroup.
Denote by $\Gamma_\mathrm{div}$ its division group. A recent conjecture due to
R\'emond, related to the Zilber-Pink conjecture, predicts that the absolute
logarithmic Weil height of an element of
$\mathbb{Q}(\Gamma_\mathrm{div})^*\backslash \Gamma_{\mathrm{div}}$ is bounded
from below by a positive constant depending only on $\Gamma$. In this paper, we
propose a new way to tackle this problem.
