2-local homotopy equivalence between G2/SO(4) and J^8

Establish a 2-local homotopy equivalence between the homogeneous space G2/SO(4) and the 8-dimensional Poincaré duality complex J^8.

Background

After proving that H*(G2/SO(4); F2) is isomorphic to H*(J^8; F2) as unstable A-algebras, the authors propose that the spaces are equivalent at the prime 2.

Confirming this conjecture would identify J^8 up to 2-local homotopy type with a well-known homogeneous space, providing a manifold model for the constructed Poincaré duality complex at the prime 2.