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Fun!!! Noah Kravitz and Mitchell Lee (arxiv.org/pdf/2307.05875.pdf) proved it in all dimensions for domains that are sufficiently close to the ball in a suitable sense so it seems like the truth is going to be complicated. Looking forward to the proof in 2D!!
Not entirely sure how related this is: there is an extension of the Stieltjes result to the six classical orthogonal polynomials that has a 1/r interaction between roots (arxiv.org/pdf/1804.09697.pdf). The induced energy can be used to quickly find roots of the six classical orthogonal polynomials by solving a coupled system of ODEs (faster methods exist, of course).
The unit ball is naturally special and stronger things can be said: there, the inequality is actually true for all subharmonic functions with constant 1 (this follows relatively quickly from the maximum principle and the mean value theorem for harmonic functions). As for the ellipse: I don't think this is likely to be a counterexample, the level sets {|x| = c} get shorter when c gets large while the boundary measure stays roughly constant.
I first thought that since all the roots have integer coordinates, the example must be somewhat robust -- after playing with it a little, it seems like it is actually fairly delicate. How was it constructed?
Amazing!! I think there is a sign flip in the last root (-45 + 18i instead of -45 - 18i) but the picture shows it very clearly and I was able to reproduce the example numerically (I get that $g(\mbox{critical point}) = 1.001...$ for the critical point in question in question).
