The pull-back conjecture for semi-free equivariant bordism

Let $\Omega^{U,S^1}_*$ be the geometric equivariant bordism ring, let $MU^{S^1}_*$ be equivariant complex bordism, and write

$$F_* = MU_*[e_V^{-1},Z_{n,V}],\qquad \Phi_* = MU_*[e_V^{\pm 1},Z_{n,V}],$$

with $V$ ranging over irreducible representations. The natural maps form a commutative square.

Pull-back conjecture. The square

$$\begin{CD} \Omega^{U,S^1}_* @>>> F_* \\ @VVV @VVV \\ MU^{S^1}_* @>>> \Phi_* \end{CD}$$

is a pull-back.

This would give a framework for determining geometric equivariant bordism inside the known homotopical theory. The source describes it as approachable through the families filtration, but provides no resolution.