Yamada's conjecture on logarithmic capacity and reproducing kernels

Let $D$ be a bounded planar domain with analytic boundary and connectivity $n>1$. For $z e w$ in $D$, let $G_D(w,z)$ be the Green function and define the logarithmic capacity by

$$c_{\beta}(z):=\exp\lim_{w\rightarrow z}\bigl(G_D(w,z)-\log|w-z|\bigr).$$

Let $B(z)$ be the unweighted Bergman kernel on the diagonal, and let $\hat K(z)$ be the conjugate Hardy $H^2$ kernel on the diagonal. Yamada's conjecture. For every $z\in D$,

$$c_{\beta}(z)^2<\pi B(z)<\hat K(z).$$

The second inequality is Saitoh's conjecture and was proved by Guan, while the strict comparison with logarithmic capacity is the remaining part of this proposed chain of inequalities.