Real models for the framed little $n$-disks operad

The little $n$-disks operad $D_n$ is a topological operad of rectilinear embeddings of a number of disjoint "little" disks into the unit disk. Its real homotopy type is known: Due to work of Kontsevich it is formal (over $\mathbb{R}$), i.e., there is a zigzag of (homotopy) Hopf cooperads relating the cooperad of differential forms $\Omega(D_n)$ with the cohomology cooperad $H(D_n)$. The framed little $n$-disks operad $fD_n$ is a generalization of $D_n$ in which one allows the little disks to be rotated. It is known to be formal over $\mathbb{R}$ for $n=2$ due to work of Giansiracusa-Salvatore and Severa. We describe the real homotopy type of $fD_n$ for higher $n$. Concretely, we show that $\Omega(fD_n)$ is quasi-isomorphic to $H(fD_n)$ (only) for n even, and quasi-isomorphic to an explicitly described combinatorial Hopf cooperad for $n$ odd.
In particular $fD_n (n>=2)$ is formal over $\mathbb{R}$ iff $n$ is even.