Shimura's conjecture on the algebraic independence of period symbols

Let $K$) be a CM field of degree $2n$, and let $\{\sigma_i\}_{i=1}^n$ be a CM type of $K$. The associated Shimura period symbols $p_K(\sigma_i,\operatorname{id})$ are elements of $\mathbb{C}^{\times}/\overline{\mathbb{Q}}^{\times}$. Shimura's conjecture. The numbers

$$p_K(\sigma_1,\operatorname{id}),\ldots,p_K(\sigma_n,\operatorname{id})$$

are algebraically independent over $\overline{\mathbb{Q}}$. Equivalently, for every CM type, they form a generating transcendence basis of the field $E_K$ generated by the period symbols over $\overline{\mathbb{Q}}$.