The formal Knizhnik-Zamolodchikov associator conjecture

Let $A(M^{\delta}_0)$ be the cooperadic algebra of Brown's compactified moduli spaces, let $\boldsymbol{\mathcal{Z}}_{\scriptstyle{\mathrm{ns}}}(A(M^{\delta}_0))$ be the associated augmented commutative algebra, and let $I(k_1,\ldots,k_r)$ denote the formal weight corresponding under evaluation to the multiple zeta value $\zeta(k_1,\ldots,k_r)$. Write $x_0,x_1$ for the associator variables and $Q$ for the indecomposable quotient. Formal associator conjecture. There is a formal version

$$\Phi\in \boldsymbol{\mathcal{Z}}_{\scriptstyle{\mathrm{ns}}}(A(M^{\delta}_0))\langle\langle x_0,x_1\rangle\rangle$$

of the Knizhnik-Zamolodchikov Drinfel'd associator, with the $I(k_1,\ldots,k_r)$ as coefficients rather than actual multiple zeta values. Consequently it defines a morphism

$$\mathfrak{grt}_1'\oplus\mathbb{Q}x^{01}\longrightarrow Q\boldsymbol{\mathcal{Z}}_{\scriptstyle{\mathrm{ns}}}(A(M^{\delta}_0)).$$

The conjecture is intended to transfer the associator's relations to the formal weights; in particular, the cited result would imply the double-shuffle relations for the $I(k_1,\ldots,k_r)$.