Homotopy type: homogeneous space versus J^8

Determine whether the homogeneous space G2/SO(4), whose mod 2 cohomology ring is isomorphic to H*(J^8; F2) as an unstable algebra over the Steenrod algebra, has the same homotopy type as the Poincaré duality complex J^8.

Background

Using results of Borel and of Borel–Hirzebruch, the authors identify a homogeneous space, G2/SO(4), whose mod 2 cohomology as an unstable Steenrod algebra agrees with that of J^8.

While cohomology matches, the authors explicitly note that it is not clear whether this homogeneous space is homotopy equivalent to J^8. Later, they formulate a 2-local version of this as a conjecture.