Higher-dimensional framed operadic formality conjecture

Let $M$ be a formal, simply connected, oriented closed $n$-manifold. Let $\mathtt{fe}_{n}$ be the framed little $n$-disks operad, let $\mathtt{f}\mathtt{G}_{H^{*}(M)}$ be the framed model built from $H^{*}(M)$, and let $\mathtt{f}\mathtt{e}_{n}^{\vee}$ denote the dual framed operadic structure. Let $\mathtt{f}\mathtt{FM}_{M}$ and $\mathtt{f}\mathtt{FM}_{n}$ be the framed Fulton–MacPherson module and operad.

Higher-dimensional framed compatibility conjecture. If the framed little $n$-disks operad $\mathtt{fe}_{n}$ is formal, then the pair $(\mathtt{f}\mathtt{G}_{H^{*}(M)},\mathtt{f}\mathtt{e}_{n}^{\vee})$ is quasi-isomorphic to the pair $(\Omega_{\mathrm{PA}}^{*}(\mathtt{f}\mathtt{FM}_{M}),\Omega_{\mathrm{PA}}^{*}(\mathtt{f}\mathtt{FM}_{n}))$.

This is a conditional extension of the operadic model to higher dimensions. The source explicitly notes that formality of the framed little $n$-disks operad is open in general, and describes the claim as closer to wishful thinking than to a conjecture.