Equivariant scissors congruence spectrum conjecture

Let $G$ be a finite group. The equivariant $SK^G$-groups, as $H$ varies over the subgroups of $G$, form a Mackey functor, and let $K_G(\underline{\mathbb{Z}})$ denote the genuine $G$-spectrum whose equivariant Euler characteristic is valued in the Burnside ring. Equivariant scissors congruence spectrum conjecture. The $\mathrm{SK}^G$ Mackey functor is $\pi_0$ of a genuine $G$-spectrum $K^\square_G(G$-\mathrm{Mfld_n^{\partial}})$, and there is a map of genuine$G$-spectra

K^\square_G(G$-\mathrm{Mfld_n^{\partial}})\to K_G(\underline{\mathbb{Z}}),

which lifts the equivariant Euler characteristic. This conjectures a genuine equivariant refinement of the scissors congruence $K$-theory spectrum and its Euler characteristic map; the source notes that the construction was subsequently announced in a preprint, but does not establish it here.