Cyclotomic expansion conjecture for orbit pure braid groups

Let $p:\tilde M\rightarrow M$ be a regular finite covering associated to a cofinite subgroup $H$ of $\pi_1(M)$, let $G_H=\pi_1(M)/H$, and let $P_{n,H}(M)$ be the fundamental group of the orbit configuration space. Let $I_{P_{n,H}(M)}$ be the augmentation ideal of $\mathbb K[P_{n,H}(M)]$, and define

$$\operatorname{gr} \mathbb K[P_{n,H}(M)]=\prod_{k\geq 0} I_{P_{n,H}(M)}^{k}/I_{P_{n,H}(M)}^{k+1}.$$

Cyclotomic expansion conjecture. There exists a filtration-preserving algebra morphism

$$\mathbb K[P_n(M)]\longrightarrow \operatorname{gr} \mathbb K[P_{n,H}(M)]\rtimes G_H^n$$

whose associated graded is the identity.

This conjecture proposes a graded expansion of the group algebra of the pure braid group in the solid-torus or covering-space setting, extending the relationship between finite-type invariants and augmentation filtrations. Its resolution status is not established by the supplied source information.