Bannai et al.'s classification conjecture for spherical 3-distance 5-designs

Let $C\subset\mathbb{S}^{n-1}$ be a spherical 3-distance 5-design.

Bannai et al.'s classification conjecture. One of the following holds: (1) $n=2$ and $C$ is a regular hexagon or a regular heptagon; (2) $C$ is a tight spherical 5-design; or (3) $C$ is derived from a tight spherical 7-design.

This conjecture asks for the classification of spherical 3-distance 5-designs. The paper rules out certain cases and reports verification in all dimensions $n\leq 1000$, but does not establish the classification in general.