David's conjecture on Lehmer-type lower bounds for abelian varieties

Let $A/K$ be an abelian variety over a number field, let $\mathcal{L}$ be an ample symmetric line bundle on $A$, and let $g=\dim A$. For $P\in A(\overline{K})$ that is not torsion under $\operatorname{End}(A_{\overline{K}})$, set

$$D_{\mathrm{tors}}=[K(A_{\mathrm{tors}},P):K(A_{\mathrm{tors}})].$$

David's conjecture. For every $\varepsilon>0$, there exists a constant $c(A/K,\mathcal{L})>0$ such that

$$\widehat{h}_{\mathcal{L}}(P)\geq\frac{c(A/K,\mathcal{L})}{D_{\mathrm{tors}}^{\frac{1}{g}+\varepsilon}}.$$

The paper notes that its main theorem answers this conjecture for elliptic curves with complex multiplication; the general statement is not resolved here.