Optimality of the D4 root system (regular 24-cell) as a 24-point spherical code in R4

Determine whether the spherical code consisting of the 24 vertices of the regular 24‑cell (equivalently, the D4 root system) in R4 minimizes the maximal inner product among all 24‑point subsets of S^3, i.e., whether it is an optimal spherical code.

Background

For k = 1 in the Kerdock construction, the spherical code is the D4 root system (the vertices of the regular 24‑cell). While this configuration solves the 4‑dimensional kissing problem, its optimality as a spherical code (maximizing minimal angle among 24 points on S^3) is not established.

The authors note three-point bounds do not seem sharp in this case and explicitly state the optimality question remains unresolved.