The relative Lehmer conjecture for tori and abelian varieties

Let $G$ be a torus or an abelian variety over a number field $K$, let $g=\dim G$, and let $K_{\mathrm{tors}}$ be the extension of $K$ generated by the coordinates of the torsion points of $G$. Let $\hat h$ be the canonical height associated with an ample line bundle on a compactification of $G$.

Relative 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

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

This conjecture replaces the degree over $\mathbf Q$ by the corresponding degree over the field generated by torsion points. The source notes that such relative Lehmer-type bounds are motivated by results over maximal cyclotomic extensions and remain conjectural in the stated generality.