One-dimensional de Rham–Betti structure conjecture for abelian varieties

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. Write $\langle\mathrm{H}^1_{\mathrm{dRB}}(A,\mathbb{Q})\rangle^{\otimes}$ for the tensor category generated by its first de Rham–Betti cohomology object, and let $\mathbb{Q}_{\mathrm{dRB}}(k)$ denote the $k$th de Rham–Betti Tate object. One-dimensional de Rham–Betti structure conjecture. Every one-dimensional object in $\langle\mathrm{H}^1_{\mathrm{dRB}}(A,\mathbb{Q})\rangle^{\otimes}$ is of the form $\mathbb{Q}_{\mathrm{dRB}}(k)$ for some $k\in\mathbb{Z}$. This conjecture concerns the one-dimensional de Rham–Betti structures arising tensorially from the cohomology of an abelian variety; the source gives no resolution status.