Positivity conjecture for Faber-polynomial error norms at corners

Let $L$ be a piecewise analytic Jordan curve, let $G$ be its bounded interior, and let $\Omega$ be its unbounded exterior. For each integer $n\geq 0$, let $\phi$ be the conformal map from $\Omega$ onto the exterior of the unit disk, normalized at infinity, let $F_n$ be the polynomial part of the Laurent expansion at infinity of $\phi^n$, and define

$$E_n(z):=\phi^n(z)-F_n(z),\qquad z\in\Omega.$$

Positivity conjecture for Faber-polynomial error norms. If $L$ has at least one corner with exterior angle different from $0$, $\pi$, and $2\pi$, then

$$\lim_{n\to\infty}\|E'_{n+1}\|^2_{L^2(\Omega)}>0.$$

The conjecture proposes that any such corner forces a nonzero limiting contribution from the derivatives of the Faber-polynomial remainders. The stated example of a piecewise analytic curve with corners supports this expectation, but no general resolution is supplied here.