Diophantine exponent conjecture for abelian varieties

Let $\mathcal{A}$ be a simple abelian variety of dimension $g$ over a number field $K$ with a fixed real embedding, and let $r$ denote the rank of the Mordell--Weil group $\mathcal{A}(K)$. For $P\in\mathcal{A}(\mathbb{R})^0\setminus\mathcal{A}(K)$, define its Diophantine exponent by

$$\sigma(P)=\limsup_{\hat{h}(Q)\to\infty}\frac{-\log\operatorname{dist}(P,Q)}{\log\hat{h}(Q)},$$

where $Q\in\mathcal{A}(K)$, and define

$$\sigma_{\mathcal{A},K}=\inf\left\{\sigma(P):P\in\mathcal{A}(\mathbb{R})^0\setminus\mathcal{A}(K)\right\}.$$

The Diophantine exponent conjecture.

$$\sigma_{\mathcal{A},K}=\frac{r}{2g}.$$

The paper presents this as a conjecture related to Waldschmidt's density conjecture and proves only an upper bound in the surrounding discussion; the equality remains open in the supplied text.