The folklore conjecture on spherical 4-distance 7-designs

Let $\mathbb{S}^{n-1}$ be the unit sphere in $\mathbb{R}^n$. A finite set $C\subset\mathbb{S}^{n-1}$ is a spherical $\tau$-design if its normalized spherical average agrees with the average over $C$ for every polynomial of degree at most $\tau$. It is a 4-distance set when the set of distinct inner products between distinct points has cardinality $4$, and it is tight when it attains the Delsarte–Goethals–Seidel bound for spherical designs.

Folklore conjecture. Let $C\subset\mathbb{S}^{n-1}$, with $n\geq 2$, be a spherical 4-distance 7-design. Then $C$ is a tight spherical 7-design. In particular,

$$|C|=2{n+2\choose 3}.$$

The paper establishes divisibility properties for the cardinality and dimension of such designs, providing a basis for computational investigation of this conjecture. The conjecture is presented as folklore whose explicit formulation the authors had not seen, and its resolution is not given here.