The stronger twisted coalgebra conjecture in right $\mathbb{E}_d$-modules

Let $M$ be a simply connected parallelizable manifold of finite type and dimension $d$. Let $\operatorname{ch}(\mathsf{FM}_M)$, $M^+$, $\mathcal{I}$, $\operatorname{Tw}$, and $\operatorname{B}_\iota$ be as above, and let $\mathbb{E}_d$ be a model of the little-$d$-disks operad. The source gives a zig-zag of quasi-isomorphisms from $\operatorname{ch}(\mathsf{FM}_M)$ to

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

Stronger twisted coalgebra conjecture. For some model of the $\mathbb{E}_d$ operad, 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 $\mathbb{E}_d$-modules.

This strengthens the preceding conjecture by requiring a full right $\mathbb{E}_d$-module structure rather than only the right $\operatorname{\Omega B}(\mathsf{Lie}_d\otimes\mathcal{E})$-module structure. The source presents it conditionally on constructing that extension and gives no resolution.