Rémond's Lehmer conjecture for tori and abelian varieties

Let $K$ be a number field, and let $G$ be either $\mathbb{G}_m^t$ for some $t\in\mathbb{Z}_{\geq 0}$ or an abelian variety over $K$. Let $\Gamma\subset G(\overline{K})$ be a subgroup of finite rank, and let $\Gamma_{\mathrm{sat}}$ be its saturated closure. For a finite extension $L/K(\Gamma_{\mathrm{sat}})$, let $h_G$ denote the canonical height specified in the source. Rémond's conjecture. For every such $L$, there exists $c_L>0$ such that

$$h_G(P)\geq c_L$$

for every $P\in G(L)\setminus\Gamma_{\mathrm{sat}}$. This is a special case of Rémond's general conjecture and is presented as the main motivation for the paper; its status is not resolved in the supplied text.