Cohen–Moore–Neisendorfer–Gray–Mahowald conjecture on charming maps and Anick spaces

Let $p$ be a prime, let $P^k(p)=S^{k-1}\cup_p e^k$ be the mod $p$ Moore space with top cell in dimension $k$, and let a charming map mean a map of the types specified in the conjecture. Let $\mathcal{Q}_1$ denote the relevant class of spaces, and let $K_2$ and $K_3$ denote the homotopy fibers of the stated integral maps.

Charming-map conjecture. The following statements are true:

1. The homotopy fiber of any charming map is equivalent as a loop space to the loop space on an Anick space.
2. There exists a $p$-local charming map

$$f:\Omega^2 S^{2p^n+1}\to S^{2p^n-1}$$

whose homotopy fiber admits a $\mathcal{Q}_1$-space retraction off of $\Omega^2P^{2p^n+1}(p)$. There are also integrally defined maps $\Omega^2S^9\to S^7$ and $\Omega^2S^{17}\to S^{15}$ whose composites with the double suspensions on $S^7$ and $S^{15}$, respectively, are the degree $2$ maps; their homotopy fibers $K_2$ and $K_3$, respectively, admit deloopings, and the homotopy fibers admit $\mathcal{Q}_1$-space retractions off $\Omega^2P^9(2)$ and $\Omega^2P^{17}(2)$, respectively.

This conjecture amalgamates conjectures attributed to Cohen, Moore, Neisendorfer, Gray, and Mahowald in unstable homotopy theory, together with an analogue of the Selick–Theriault rigidity result. The first assertion would follow from the earlier conjectural equivalence involving $BW_n$ and an Anick space if the Cohen–Moore–Neisendorfer map were a Gray map; the status of the stated assertions is not resolved in the supplied source.