The twisted coalgebra conjecture for configuration-space models

Let $M$ be a simply connected parallelizable manifold of finite type and dimension $d$. Let $\operatorname{ch}(\mathsf{FM}_M)$ be the symmetric sequence of chains on the Fulton–MacPherson compactifications, let $M^+$ be the one-point compactification, let $\mathcal{I}$ be the unit operad, and let $\operatorname{Tw}$ and $\operatorname{B}_\iota$ denote the twist and bar constructions. The source provides a zig-zag of quasi-isomorphisms identifying $\operatorname{ch}(\mathsf{FM}_M)$ aritywise with

$$\operatorname{B}_\iota\bigl(\operatorname{ch}^*(M^+)\otimes\operatorname{\Omega B}(\mathsf{sLie}\otimes\mathcal{E})\circ(s^d\mathcal{I})\bigr).$$

Twisted coalgebra conjecture. This zig-zag can be promoted to a zig-zag of equivalences

$$\operatorname{Tw}(\operatorname{ch}(\mathsf{FM}_M))\simeq\operatorname{B}_\iota\bigl(\operatorname{ch}^*(M^+)\otimes\operatorname{\Omega B}(\mathsf{sLie}\otimes\mathcal{E})\circ(s^d\mathcal{I})\bigr),$$

of twisted $\operatorname{\Omega B}\mathcal{E}^{\mathrm{nu}}$-coalgebras in right $\operatorname{\Omega B}(\mathsf{Lie}_d\otimes\mathcal{E})$-modules.

The conjecture is motivated by the relation between configuration spaces and $p$-adic homotopy types. The source notes that it holds in characteristic zero and gives supporting evidence from the $d$-shifted Lie-module structure, but leaves the positive-characteristic assertion open.