Characterisation of Cayley hypersurfaces by affine symmetry and parallel normals

Let $\Sigma$ be a non-degenerate hypersurface in affine $N$-space. Assume that $\Sigma$ admits a transitive Abelian group of affine motions, that its full symmetry group has one-dimensional isotropy, and that its affine normals are everywhere parallel; equivalently, $\Sigma$ is an improper affine hypersphere.

Cayley hypersurface characterisation conjecture. Then, in a suitable affine coordinate system, $\Sigma$ is given by the Cayley hypersurface equation.

The conjecture proposes that these three geometric properties characterise the Cayley hypersurfaces among non-degenerate affine hypersurfaces. The supplied text does not state whether the conjecture has been proved or disproved.