Rognes's equivariant common basis complex equivalence conjecture

Let $R$ be a ring, let $k$ be a positive integer, let $D^V(R^k)$ denote the relevant decomposition space, and let $\P^\circ$ be the poset of nontrivial minimal spanning posets, with nerve $N\P^\circ$. The group $GL_k(R)$ acts naturally on these spaces.

Rognes's equivariant equivalence conjecture. There should be a $GL_k(R)$-equivariant equivalence

$$\Sigma^{-1}D^V(R^k)\simeq\Sigma N\P^\circ.$$

After applying $\Sigma^\infty$, the equivalence is known, so the spaces have the same homology. For $k\geq2$ both spaces are simply connected; the remaining issue described in the source is to construct an equivariant map inducing an isomorphism on homology.