Equivariant Ching conjecture for the little 2-disks operad

Let $D_2$ be the little $2$-disks operad equipped with its $S^1$-action, let $\Sigma^\infty_+$ denote suspension spectrum after adjoining a disjoint basepoint, and let $K$ be the derived Koszul dual functor on operads in spectra. Equivariant Ching conjecture. The operad $K(\Sigma^\infty_+D_2)$ is equivalent to a suitable operadic suspension of $\Sigma^\infty_+D_2$ in the category of $S^1$-spectra. This strengthens Ching's corresponding nonequivariant conjecture in arity dimension $n=2$; the source gives equivariant formality results as evidence but does not state that the conjecture is resolved.