Semi-parallelism and parallelism conjecture for locally strongly convex affine hyperspheres

Let $M$ be a locally strongly convex affine hypersphere of dimension $n$. Its cubic form $C$ is semi-parallel when $hat{R}\cdot C=0$ and parallel when $hat{\nabla}C=0$, where $hat{R}$ and $hat{\nabla}$ are respectively the curvature tensor and Levi-Civita connection of the affine metric.

Semi-parallelism and parallelism conjecture. The two conditions are equivalent:

$$\hat{R}\cdot C=0\quad\Longleftrightarrow\quad\hat{\nabla}C=0.$$

Parallel cubic form always implies semi-parallel cubic form, but the converse is not generally true for affine hypersurfaces. The conjecture asks whether the converse holds for every locally strongly convex affine hypersphere; it extends classification results known in lower-dimensional and special cases.