Rémond's Bogomolov property conjecture for radical extensions

Let $\mathbb{K}$ be a number field, and let $\tilde\Gamma\subseteq\overline{\mathbb{Q}}^\times$ be a subgroup of finite rank. Write $\tilde\Gamma^{\mathrm{div}}$ for its divisible hull, consisting of the elements whose some positive integral power lies in $\tilde\Gamma$. For a subset $E$ of $\overline{\mathbb{Q}}$, say that $E$ has the Bogomolov property if its elements outside the roots of unity have Weil height bounded below by a positive constant. Rémond's conjecture. The set

$$\mathbb{K}(\tilde\Gamma)\setminus\tilde\Gamma^{\mathrm{div}}$$

has the Bogomolov property. This conjecture places earlier results on small points in radical extensions in a broader framework; the source states that it is a special case of Rémond's Conjecture 3.4, and notes that related partial results were known. The stated case was recently proved by S. Checcoli and G. Dill, so its status is solved.