On the compatibility of the Betti harmonic coproduct with cyclotomic filtrations

In a previous paper, the second author introduced a Betti counterpart of $N$-cyclotomic double shuffle theory for any $N \geq 1$. The construction is based on the group algebra of the free group $F_2$, endowed with a filtration relative to a morphism $F_2 \to \mu_N$ (where $\mu_N$ is the group of $N$-th roots of unity). One of the main results therein is the construction of a complete Hopf algebra coproduct $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ on the relative completion of a specific subalgebra $\mathcal{W}^\mathrm{B}$ of the group algebra of $F_2$. However, an explicit formula for this coproduct is missing. In this paper, we show that the discrete Betti harmonic coproduct $\Delta^{\mathcal{W}, \mathrm{B}}$ defined in \cite{EF1} for the classical case ($N=1$) by the first author and Furusho remains compatible with the filtration structure on $\mathcal{W}^\mathrm{B}$ induced by the relative completion for arbitrary $N$. This compatibility suggests that the completion corresponding to $\Delta^{\mathcal{W}, \mathrm{B}}$ is a candidate for an explicit realization of $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$.