Additivity and equivariant little disk embeddings

Given a smooth manifold with action of a finite group $G$ and an equivariant tangential structure $\Theta$, the spaces of $\Theta$-framed equivariant disk embeddings form a "$\Theta$-disk $G$-presheaf" on the underlying $G$-category of a $G$-symmetric monoidal category of disks in $\Theta$-framed orthogonal representations. This occurs as the $G$-symmetric envelope of a $G$-operad, called the "$\Theta$-framed little disks $G$-operad" $\mathbb{E}_\Theta$, i.e. $\Theta$-disk $G$-presheaves should be thought of as right $\mathbb{E}_\Theta$-modules. In this talk, I'll sketch an extension of Dunn-Lurie additivity to these $G$-operads. In principle, we reconstruct the $\Theta \oplus \Psi$-disk $G$-presheaves associated with products of $\Theta$- and $\Psi$-framed $G$-manifolds from their factors.