The homogeneous-space p-adic Grothendieck period conjecture

Let $M$ be a motive with good reduction and rational coefficients, unramified above $p$. Let $\mathcal H_M$ be the associated space of motivic $p$-adic periods, and let $\mathcal P_p(\langle M\rangle)$ be the algebra generated by the $p$-adic periods of all objects in the tannakian category generated by $M$.

Homogeneous-space p-adic Grothendieck period conjecture. The variety $\mathcal H_M$ is irreducible and

$$\operatorname{trdeg}_{\overline{\mathbb Q}}\bigl(\mathcal P_p(\langle M\rangle)\bigr)=\dim(\mathcal H_M).$$

This is the other strong formulation of the $p$-adic Grothendieck period conjecture. It concerns the full tensor-generated period algebra and the motivic homogeneous space; its equivalence with the tensorial formulation is established in the paper, but the conjecture itself remains open.