Rémond's height conjecture for division groups

Let $Gamma oindent ot i$ be a finitely generated subgroup of $Q^*$, and let $Gamma_{\mathrm{div}}$ be its division group. For $alpha\in\overline{\mathbb{Q}}^*$, write $h(\alpha)$ for its absolute logarithmic Weil height.

Rémond's conjecture. The following forms are expected to hold:

1. There is a positive constant $c$, depending only on $\Gamma$, such that

$$h(\alpha)\geq \frac{c}{[\mathbb{Q}(\Gamma_{\mathrm{div}},\alpha):\mathbb{Q}(\Gamma_{\mathrm{div}})]}$$

for all $\alpha\in\overline{\mathbb{Q}}^*\setminus\Gamma_{\mathrm{div}}$.

2. For every $\varepsilon>0$, there is a positive constant $c_\varepsilon$, depending only on $\Gamma$ and $\varepsilon$, such that

$$h(\alpha)\geq \frac{c_\varepsilon}{[\mathbb{Q}(\Gamma_{\mathrm{div}},\alpha):\mathbb{Q}(\Gamma_{\mathrm{div}})]^{1+\varepsilon}}$$

for all $\alpha\in\overline{\mathbb{Q}}^*\setminus\Gamma_{\mathrm{div}}$.

3. Small-height elements of $\mathbb{Q}(\Gamma_{\mathrm{div}})^*$ belong to $\Gamma_{\mathrm{div}}$.