  We prove that the D_4 root system (equivalently, the set of vertices of the
regular 24-cell) is not a universally optimal spherical code. We further
conjecture that there is no universally optimal spherical code of 24 points in
S^3, based on numerical computations suggesting that every 5-design consisting
of 24 points in S^3 is in a 3-parameter family (which we describe explicitly,
based on a construction due to Sali) of deformations of the D_4 root system.
