The abelian Lehmer problem

Let $A/K$ be an abelian variety over a number field and let $\mathcal{L}$ be a symmetric ample line bundle on $A$. For $P\in A(\overline{K})$, let $\delta_{\mathcal{L},K}(P)$ denote the obstruction index relative to $\mathcal{L}$ and $K$, and let $g_0$ be the dimension of the smallest algebraic subgroup containing $P$.

Abelian Lehmer problem. There is a constant $c(A/K,\mathcal{L})>0$ such that, for every point $P\in A(\overline{K})$ of infinite order modulo every proper abelian subvariety of $A$,

$$\widehat{h}_{\mathcal{L}}(P)\geq\frac{c(A/K,\mathcal{L})}{\delta_{\mathcal{L},K}(P)}.$$

Moreover, for every nontorsion point $P\in A(\overline{K})$, with $D=[K(P):K]$,

$$\widehat{h}_{\mathcal{L}}(P)\geq\frac{c(A/K,\mathcal{L})}{D^{1/g_0}}.$$

This is a Lehmer-type lower bound for canonical heights on abelian varieties with complex multiplication. The source presents it as a problem and gives no resolution.