Saitoh's strict inequality conjecture for conjugate Hardy and Bergman kernels

Let $D$ be a planar region bounded by finite analytic Jordan curves, let $z_0\in D$, and let $B_D(z_0)$ denote the Bergman kernel on $D$. Let $\hat K_D(z_0)$ be the conjugate Hardy $H^2$ kernel, defined by

$$\hat K_D(z_0):=\frac{1}{\inf\left\{\|f\|^2_{H^2_{(c)}(D)}:f\in H^2_{(c)}(D)\ \&\ f(z_0)=1\right\}}.$$

Saitoh's conjecture. If $D$ is not simply connected, then

$$\hat K_D(z_0)>\pi B_D(z_0).$$

This conjecture compares the conjugate Hardy $H^2$ kernel with the Bergman kernel and predicts strict inequality for multiply connected planar regions. Its resolution status is not specified in the supplied text.