Rémond's Dobrowolski conjecture for finite-rank subgroups

Let $K$ be a number field, let $G$ be a semiabelian variety over $K$, and let $\Gamma\subset G(\overline{K})$ be a subgroup of finite rank. For a subvariety $V\subset G$, write $\mu_{\mathrm{ess}}(V)$ for its essential minimum and $\omega_{K(\Gamma)}(V)$ for its obstruction index relative to $K(\Gamma)$. Call $\Gamma$ a Dobrowolski group if, for every $\varepsilon>0$, there exists $c_\Gamma(\varepsilon)>0$ such that

$$\omega_{K(\Gamma)}(V)^{1+\varepsilon}\mu_{\mathrm{ess}}(V)\geq c_\Gamma(\varepsilon)$$

for every $\Gamma$-transverse subvariety $V\subset G$. Rémond's Dobrowolski conjecture. Every subgroup of $G(\overline{K})$ of finite rank is a Dobrowolski group. This is stated as a weaker form of Rémond's Lehmer conjecture and is sufficient for the special case pursued in the paper; the supplied text does not state that it is solved.