Conjectural spectrum-level splittings for the unoriented Thom spectra

Let $Mj_2$, $Mj_3$ and $Mj^{\mathrm{c}}$ denote the indicated $\mathcal{E}_\infty$ Thom spectra, and let $k\mathrm{O}$, $\mathrm{tmf}$ and $k\mathrm{U}$ denote connective real K-theory, connective topological modular forms, and connective complex K-theory, respectively. A wedge of module spectra means a coproduct in spectra whose summands are modules over the specified ring spectrum.

Spectrum-level splitting conjecture. As a spectrum, $Mj_2$ is a wedge of $k\mathrm{O}$-module spectra, $Mj_3$ is a wedge of $\mathrm{tmf}$-module spectra and $Mj^{\mathrm{c}}$ is a wedge of $k\mathrm{U}$-module spectra.

These proposed splittings are supported by the algebraic splittings of the homology comodules. The cases involving $Mj_1$ are known, while the asserted spectrum-level splittings in the displayed cases remain conjectural.