David-Hindry's multihomogeneous abelian Lehmer conjecture

Let $A/K$ be an abelian variety of dimension $g$ over a number field and let $\mathcal{L}$ be a symmetric ample line bundle on $A$. For an $n$-tuple $(P_1,\ldots,P_n)$, write $D=[K(P_1,\ldots,P_n):K]$.

David-Hindry's conjecture. For every integer $n\in\mathbb{N}$, there is a constant $c(A/K,\mathcal{L},n)>0$ such that, for every $n$-tuple $(P_1,\ldots,P_n)$ of points of infinite order in $A(\overline{K})$ that are linearly independent over $\operatorname{End}(A)$,

$$\prod_{i=1}^n\widehat{h}_{\mathcal{L}}(P_i)\geq\frac{c(A/K,\mathcal{L},n)}{D^{1/g}}.$$

This is the multihomogeneous version of the abelian Lehmer problem, stated in the source as Conjecture 1.6 of David and Hindry. The source gives no resolution.