The Lehmer-type height bound for points on group varieties

Let $G$ be a group variety defined over a number field $k$. Let $b1$ be a point of $G$, and let $B$ be the irreducible torsion variety of minimal dimension containing $b1$. The field $k_\mathrm{tor}$ is the field of definition of the torsion of $G$. Lehmer-type bound. For every real $\eta>0$, there exists a positive constant $c(G,\eta)>0$ such that

$$h(\alpha)\ge c(G,\eta)\frac{(\deg B)^{\frac{1}{\dim B}-\eta}}{[k_\mathrm{tor}(\alpha):k_\mathrm{tor}]^{\frac{1}{\dim B}+\eta}}.$$

This generalizes Lehmer's conjecture and is known in several toric and special abelian-variety cases, but no method is known for such a sharp bound in a general abelian variety.