David's Lehmer conjecture for tori and abelian varieties

Let $G$ be a torus or an abelian variety over $\overline{\mathbf Q}$, let $g=\dim G$, and let $\hat h$ be a canonical height associated with an ample line bundle on a compactification of $G$.

David's Lehmer conjecture. There is a real number $c>0$ such that every point $x\in G(\overline{\mathbf Q})$ not contained in any proper algebraic subgroup of $G$ satisfies

$$\hat h(x)\geq c\,[\mathbf Q(x):\mathbf Q]^{-1/g}.$$

This is a higher-dimensional Lehmer-type lower bound for canonical heights. The source explains that the bound is known up to an arbitrarily small loss in the exponent for tori and abelian varieties with complex multiplication, while the general statement remains open.