The no-maximizer conjecture for Bergman N-polynomial content

Let $\Omega$ be a simply connected Jordan region in the plane. For $N\in\mathbb N$, let $\rho_N(\Omega)$ denote the squared distance from $\bar z$ to the polynomials of degree at most $N$ in $L^2(\Omega,dA)$. No-maximizer conjecture. For each $N\in\mathbb N$, there is an $n\in\mathbb N$ such that among all $n$-gons with area $1$, $\rho_N$ has no maximizer. The conjecture asserts a fundamental difference between Bergman analytic content and Bergman $N$-polynomial content; the paper gives examples and evidence but does not resolve the claim.