Normal-derivative Green-function conjecture for C-GNP domains

Let $4\Omega4$ be a domain, let $4G_{\Omega}(x,y)4$ be its Green's function, and let $4\tau_{\Omega}(x)4$ denote the thickness at a boundary point $4x4$. Suppose that $4x\in\partial\Omega4$ is a point where the normal $4\nu4$ exists.

Green-function normal-derivative conjecture. As $4y4$ approaches $4x4$ along the normal,

$$\frac{\partial G_{\Omega}}{\partial \nu}(x,y) \sim \frac{1}{\tau_{\Omega}(x)}.$$

The proposed relation is intended to quantify the singularity of the normal derivative of the Green's function near the boundary. The source gives no evidence that this potential-theoretic statement has been resolved.