Saitoh's conjecture for conjugate Hardy $H^{2}$ kernels

Let $D$ be a planar regular region whose boundary consists of $n$ analytic Jordan curves. Let $\hat{R}(z,\bar{w})$ be its conjugate Hardy $H^{2}$ kernel and $B(z,\bar{w})$ its Bergman kernel; write $\hat{R}(z)=\hat{R}(z,\bar{z})$ and $B(z)=B(z,\bar{z})$. Saitoh's conjecture. If $n>1$, then

$$\hat{R}(z)>\pi B(z).$$

The conjecture compares the diagonal conjugate Hardy $H^{2}$ and Bergman kernels on multiply connected planar regions. The paper proves this strict inequality, resolving the conjecture.