The sparse-presentation conjecture for the J-induced map on the Nakaoka module

Let the Nakaoka module be presented using generators including $\gamma_{1[1]}^{2^k-1}$, and consider the map induced by the $J$-homomorphism. Sparse-presentation conjecture. There exists a presentation of the Nakaoka module such that every generator other than $\gamma_{1[1]}^{2^k-1}$ maps to $0$ under the map induced by the $J$-homomorphism. The paper offers this as an apparent structural explanation for the behavior seen at the level of resolutions, but does not prove the existence of such a presentation.