Relative semi-abelian Schanuel conjecture

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$, and let $G$ be an extension of $A$ by the torus $\mathbb{G}_m^s$ parametrized by $Q_1,\dots,Q_s\in A^*(\overline{\mathbb{Q}})$. Let $\Omega_G$ be its periods and let $R_1,\dots,R_n\in G(\mathbb{C})$. Relative semi-abelian conjecture.

$$\operatorname{tr.deg}_{\overline{\mathbb{Q}}(\Omega_G)}\overline{\mathbb{Q}}(\Omega_G,R_1,\dots,R_n,\widetilde{\log}_G(R_1),\dots,\widetilde{\log}_G(R_n))\geq2\dim B_Q+\dim Z(1).$$

Here $B_Q$ and $Z(1)$ are the algebraic groups defined in the paper's preceding construction. This is a relative form of the semi-abelian analogue of Schanuel's conjecture, with the transcendence degree measured over the field generated by the periods of $G$.