The Bergman N-polynomial content conjecture for convex polygons

Let $\Omega$ be a planar region, let $\rho_N(\Omega)$ be the squared distance from $\bar z$ to the polynomials of degree at most $N$ in $L^2(\Omega,dA)$, and call the corresponding distance the Bergman $N$-polynomial content. For $n,N\in\mathbb N$ with $n\geq 3$, consider convex $n$-gons of area $1$. Bergman $N$-polynomial content conjecture. The convex $n$-gon of area $1$ that maximizes the Bergman $N$-polynomial content is the regular $n$-gon. This conjecture is presented as a generalization of Pólya's conjecture, with significant numerical and structural evidence supplied in the paper.