Simple algebraic-group closure conjecture over the reals and p-adics

Let $\mathcal{A}$ be a connected commutative algebraic group over $\mathbb{Q}$, let $\mathcal{L}\subset\mathcal{A}(\mathbb{Q})$ be finitely generated, and let $\overline{\mathcal{L}}$ denote its closure in either $\mathcal{A}(\mathbb{R})$ or $\mathcal{A}(\mathbb{Q}_p)$. Suppose that $\mathcal{A}$ is simple over $\mathbb{Q}$. Simple closure conjecture. In $\mathcal{A}(\mathbb{R})$, either $\mathcal{L}$ is discrete or $\mathcal{L}$ is dense. In $\mathcal{A}(\mathbb{Q}_p)$, if $\mathcal{L}$ is contained in a maximal compact subgroup, then $\overline{\mathcal{L}}$ is, up to the stated identification, $\mathbb{Z}_p^\ell$, where $\ell=\min(\operatorname{rank}\mathcal{L},\dim\mathcal{A})$. The claim is presented as known for simple groups and for \mathcal{A}=R_{K/\mathbb{Q}(\mathbb{G}_m), but no resolution status is supplied for the conjectural assertions.