Existence of weighted 15-designs on the 3-sphere with more than five proper orbits

Determine whether there exists a weighted $15$-design on $\mathbb{S}^3$ of type (\ref{eq:invariant0}) with more than five proper orbits.

Background

The generalized corner-vector method yields a uniform upper bound of degree $15$ for the relevant BnB_n-invariant spherical designs when n≥4n\geq4. In dimension four, the paper exhibits weighted $11$-designs with three proper orbits and weighted $13$-designs with five orbits, but reports no $15$-design with more than five proper orbits and therefore formulates the existence question explicitly.

References

Does there exist a weighted $15$-design on $\mathbb{S}3$ of type (\ref{eq:invariant0}) with more than $5$ proper orbits?

— More on the corner-vector construction for spherical designs  (2501.11437 - Tanino et al., 20 Jan 2025) in Problem 4.1, Section 5, “Designs with more than two proper orbits on $\mathbb{S}^3$”