The period conjecture for motivic periods

Let $\mathcal{P}^{\rm m}$ be the algebra of motivic periods, let $\mathcal{P}$ be the algebra of periods, and let the period map be the evaluation homomorphism

$$\mathop{\rm per}\nolimits\colon \mathcal{P}^{\rm m} \longrightarrow \mathcal{P}.$$

The period conjecture. The period map is an isomorphism.

The map is explicitly surjective, whereas injectivity is not proven; injectivity expresses that a period has a unique motivic realisation.