Unstable Adams spectral sequence for approximation spheres

Define an appropriate Adams spectral sequence for the approximation spheres S^n_k that interpolates between the unstable Adams spectral sequence of S^n and the stable Adams spectral sequence of \mathbb{S}^n.

Background

The computations in the paper are carried out at the space level using stable Adams spectral sequences. The authors suggest that these calculations should lift to an Adams-type spectral sequence or an unstable synthetic category, but explicitly state that even the correct notion of such a spectral sequence is not known.