The completed finite period conjecture

Let $\cA:=\cH/\zeta^{\fm}(2)\cH$, where $\cH$ is the ring of motivic multiple zeta values, and let $\cA((T))$ be the ring of formal Laurent series over $\cA$. Let

$$\mathrm{Fil}^n\cA((T))=T^n\cA[[T]].$$

Let $\mathbb{Q}_{p\to\infty}$ be the quotient of families $(a_p)\in\prod_p\mathbb{Q}_p$ with valuations bounded below by families whose valuations tend to infinity, with filtration

$$\mathrm{Fil}^n\mathbb{Q}_{p\to\infty}=\{(a_p):\liminf v_p(a_p)\geq n\}.$$

The completed finite period map is the continuous ring homomorphism

$$\widehat{per}:\cA((T))\to\mathbb{Q}_{p\to\infty}.$$

Period conjecture. For every integer $n$,

$$\widehat{per}^{-1}(\mathrm{Fil}^n\mathbb{Q}_{p\to\infty})=\mathrm{Fil}^n\cA((T)).$$

In particular, $\widehat{per}$ is injective. This is an analogue of the Grothendieck period conjecture: it asserts that the completed finite period map detects exactly the filtrations on its source and target. The paper presents it as an expectation, and no resolution is supplied.