Relative Lehmer conjecture for heights relative to divisible subgroups

Let $g_1,\dots,g_n\in\overline{\mathbb{Q}}^\times$ and let $\Gamma=\langle g_1,\dots,g_n\rangle^{\mathrm{div}}$ be a finite-rank divisible subgroup of $\mathbb{G}_m(\overline{\mathbb{Q}})$. Define

$$h_\Gamma(\alpha)=\inf_{\gamma\in\Gamma}h(\alpha/\gamma),$$

and let $k=\mathbb{Q}(g_1,\dots,g_n)$. Relative Lehmer conjecture. For every $\varepsilon>0$, there exists $C_\Gamma(\varepsilon)>0$ such that, for every $\alpha\in\mathbb{G}_m(\overline{\mathbb{Q}})\setminus\Gamma$,

$$h_\Gamma(\alpha)\geq\frac{C_\Gamma(\varepsilon)}{[k^{\mathrm{cyc}}(\alpha):k^{\mathrm{cyc}}]^{2+\varepsilon}}.$$

This is presented as a weaker version of the weak form of Rémond's generalized Lehmer conjecture, and its resolution is not established by the supplied text.