Equivariant Milnor map conjecture

Let $T^k$ be the $k$-torus and let $\mathbb{Z}_2^k$ be its real subgroup. Write $\Omega^{U,T^k}_{2*}$ and $\Omega^{U,\mathbb{Z}_2^k}_{2*}$ for the unitary equivariant bordism groups, and $\Omega^{O,\mathbb{Z}_2^k}_{*}$ for the unoriented equivariant bordism group. Equivariant Milnor map conjecture. Whenever $G$ is either $T$ or $\mathbb{Z}_2$, there exists an equivariant Milnor map

$$\mu^{G^k}: \Omega^{U,G^k}_{2*} \to \Omega^{O,\mathbb{Z}_2^k}_{*}$$

reducing the dimension by a half, defined as the real points of smooth complex generators, which is moreover surjective. This conjecture proposes an equivariant generalization of the Milnor map and would relate unitary equivariant bordism to unoriented equivariant bordism through real points of smooth complex generators. The source presents preliminary results in this direction but does not establish the conjecture.