Suita's conjecture on the Bergman kernel and logarithmic capacity

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_{\Omega}$. For $z_0\in\Omega$, let $c_{\beta}(z_0)$ denote its logarithmic capacity and let $B_{\Omega}(z_0)$ denote the Bergman kernel function. Suita's conjecture. One has

$$\pi B_{\Omega}(z_0)\geq \bigl(c_{\beta}(z_0)\bigr)^2,$$

for every $z_0\in\Omega$, with equality if and only if $\Omega$ is conformally equivalent to the unit disc less a possible closed set of inner capacity zero. The inequality part has been proved for bounded planar domains by Blocki and for open Riemann surfaces by Guan and Zhou; the equality characterization is the remaining part addressed in this paper.