Weak abelian Schanuel conjecture

Let $A$ be an abelian variety of dimension $g$ over $\overline{\mathbb{Q}}$, with exponential map $\exp_A:\mathbb{C}^g\to A(\mathbb{C})$, period and quasi-period matrix $\widetilde{\Lambda}_A$, and vector of abelian zeta integrals $\zeta_A(u)$. Let $u\in\mathbb{C}^g$, and let $H$ be the smallest algebraic subgroup of $A$ containing $\exp_A(u)$. Write $\overline{\mathbb{Q}}(\widetilde{\Lambda}_A)$ for the field generated over $\overline{\mathbb{Q}}$ by the periods and quasi-periods.

Weak abelian Schanuel conjecture.

$$\operatorname{trdeg}_{\overline{\mathbb{Q}}(\widetilde{\Lambda}_A)}\overline{\mathbb{Q}}(\widetilde{\Lambda}_A,\exp_A(u),u,\zeta_A(u))\ge 2\dim(H).$$

This is presented as a weaker version of André's generalised period conjecture and is an abelian analogue of Schanuel's conjecture. The source does not state a resolution of this formulation, so it remains open.