The period-characterisation conjecture for geometric Eisenstein words

Let $V_{2n}^{\operatorname{dR}}$ be the $SL_2$-representation appearing above, let $\mathcal{O}(\mathcal{U}_E^{\operatorname{dR}})$ be the Hopf subalgebra generated by Eisenstein words, and let $\mathcal{O}(\mathcal{U}^{\operatorname{geom}})$ be the geometric Hopf subalgebra. For

$$f\in \operatorname{Hom}_{SL_2}\bigl(V_{2n}^{\operatorname{dR}},\mathcal{O}(\mathcal{U}_E^{\operatorname{dR}})\bigr),$$

write $\mathcal{C}_S$ for the corresponding point of $\mathcal{U}_{1,1}^{\operatorname{dR}}(\mathbb{C})=\operatorname{Hom}(\mathcal{O}(\mathcal{U}_{1,1}^{\operatorname{dR}}),\mathbb{C})$. Period-characterisation conjecture. The image of $f$ is contained in $\mathcal{O}(\mathcal{U}^{\operatorname{geom}})$ if and only if

$$\operatorname{im}(\mathcal{C}_S\circ f)\subseteq \mathcal{Z}[(2\pi i)^\pm].$$

This proposes that, among $SL_2$-equivariant maps with values in the Eisenstein-word Hopf subalgebra, belonging to the geometric subalgebra is exactly characterised by all associated periods being in the multiple-zeta-value algebra with powers of $2\pi i$ inverted. The conjecture is posed after the preceding inclusion from geometric words to $\mathcal{Z}[(2\pi i)^\pm]$; its resolution is not specified here.