Period conjecture for Galois biconjugates

Let $\xi\in\mathcal{P}^{\mathfrak{m},+}$ be a motivic period, and let $P_{\xi}\subset\mathbb C$ be the $\mathbb Q$-algebra generated by the images under the period homomorphism of the Galois biconjugates of $\xi$. Let $\dim_{\mathrm{tr}}(\xi)$ denote the transcendence dimension of $\xi$. Period conjecture for Galois biconjugates.

$$\deg_{\mathrm{tr}}P_{\xi}=\dim_{\mathrm{tr}}(\xi).$$

The source states this as a consequence of the effective period conjecture, comparing the transcendence degree of the algebra generated by the biconjugates with the motivic Galois dimension; its independent status is not specified.