Waldschmidt's density conjecture for rational points on abelian varieties

Let $\mathcal{A}$ be a simple abelian variety of dimension $g$ over a number field $K$ embedded in $\mathbb{R}$. Let $\mathcal{A}(\mathbb{R})^0$ be the identity component, let $\hat{h}$ be the canonical height, and define

$$\hat{\eta}_{\mathcal{A}}(h)=\inf\left\{\delta:\text{ for every }\zeta\in\mathcal{A}(\mathbb{R})^0\text{ there exists }\gamma\in\mathcal{A}(K)\text{ with }\hat{h}(\gamma)\leq h\text{ and }\operatorname{dist}(\zeta,\gamma)<\delta\right\}.$$

Denote by $r$ the rank of the Mordell--Weil group $\mathcal{A}(K)$. Waldschmidt's conjecture. For every $\varepsilon>0$, there exists $h_0>0$, depending only on $\mathcal{A}$, $K$, and $\varepsilon$, such that for every $h\geq h_0$,

$$\hat{\eta}_{\mathcal{A}}(h)\leq h^{-(r/2g)+\varepsilon}.$$

This conjecture predicts the optimal rate at which points of bounded canonical height from $\mathcal{A}(K)$ approximate the real identity component, with the exponent governed by the Mordell--Weil rank. The paper relates it to a matrix-approximation property, but does not establish the conjecture in general.