Saitoh's conjecture for multiply connected planar regions

Let $D$ be a planar regular region with finitely many boundary components that are analytic Jordan curves. Let $c_{\beta}(z)$ denote the logarithmic capacity and let $B(z)$ denote the Bergman kernel on $D$. If $D$ is not simply connected, Saitoh's conjecture. one has

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

This is the unweighted Saitoh conjecture comparing logarithmic capacity, the Bergman kernel, and the conjugate Hardy kernel; the supplied text does not indicate whether this claim has been resolved.