Equivalence of fixed points of Real spin bordism with MSpin

Determine whether either the genuine fixed points spectrum $(\MSpin^c_{\RR})^{C_2}$ or the homotopy fixed points spectrum $(\MSpin^c_{\RR})^{hC_2}$ of the Real spin bordism spectrum $\MSpin^c_{\RR}$ is equivalent to the spin bordism spectrum $\MSpin$.

Background

The paper constructs the Real spin bordism spectrum $\MSpin^c_{\RR}$ and shows it refines standard orientations of topological K-theories. Given that $\Spin^c(n)^{C_2}=\Spin(n)$ and $\KU_{\RR}^{C_2}=\KO$, one might expect the $C_2$-(homotopy) fixed points of $\MSpin^c_{\RR}$ to recover $\MSpin$.

However, Theorem \ref{intro.counterspin} provides an obstruction to the anticipated comparison map being an equivalence that covers $\MSO$. Despite this, the authors explicitly note that it remains unknown whether an equivalence $(\MSpin^c_{\RR})^{C_2}\simeq \MSpin$ or $(\MSpin^c_{\RR})^{hC_2}\simeq \MSpin$ exists, though they caution there is no reason to expect such an equivalence in light of the obstruction.