Grothendieck–André periods conjecture for the 1-motive $M$

Let $M=[u:\mathbb{Z}\to G^n]$ be the 1-motive with $u(1)=(R_1,\dots,R_n)$ defined by the points in the source. Let $k$ be the endomorphism field and let $\mathrm{LieBracket}$ and $\mathrm{NoLieBracket}$ denote the source's index sets. Grothendieck–André periods conjecture for $M$. The transcendence degree of the explicitly displayed period field is at least

$$\frac{4}{\dim_{\mathbb{Q}}k}+2\dim_k\langle p_i,q_j\rangle_{i,j}+\dim_{\mathbb{Q}}\langle\beta_{i,j}+\beta_{i,j}^t\rangle_{(i,j)\in\mathrm{LieBracket}}+\dim_{\mathbb{Q}}\langle t_{ij}\rangle_{(i,j)\in\mathrm{NoLieBracket}}.$$

If $\mathbb{Q}(g_2,g_3,Q_j,R_i)_{i,j}\subseteq\overline{\mathbb{Q}}$, equality should hold. This is the explicit periods statement for the paper's central family of 1-motives and underlies the equivalence with the semi-elliptic conjecture. The general assertion remains open.