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Let 
$\Gamma \subset \overline {\mathbb {Q}}^*$ be a finitely generated subgroup. Denote by 
$\Gamma _{\mathrm {div}}$ its division group. A recent conjecture due to Rémond, 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.
