Nonexistence of a genuine splitting of Real spin bordism

Let $C_2$ be the cyclic group of order two, let $\operatorname{MSpin}^c_{\operatorname{\mathbb{R}}}$ denote Real spin bordism, let $\mathcal{P}$ be the indexing set for the Anderson--Brown--Peterson splitting, let $\operatorname{ku}_{\operatorname{\mathbb{R}}}\langle 4|I|\rangle$ be the corresponding connective Real complex $K$-theory summand for $I\in\mathcal{P}$, and let $Z$ be the remaining summand. Nonexistence conjecture. There does not exist a map of genuine $C_2$-spectra

$$\operatorname{MSpin}^c_{\operatorname{\mathbb{R}}} \longrightarrow \left(\bigvee_{I\in\mathcal{P}}\operatorname{ku}_{\operatorname{\mathbb{R}}}\langle 4|I|\rangle\right)\vee Z$$

whose induced map on underlying spectra is the Anderson--Brown--Peterson map. The proposition preceding this conjecture proves nonexistence for the individual summand when $|I|$ is odd, and the authors explain that this obstructs the naive genuine refinement of the Anderson--Brown--Peterson splitting; the general assertion remains open.