For any open Riemann surface $X$ admitting Green functions, Suita asked about the precise relations between the Bergman kernel and the logarithmic capacity. It was conjectured that the Gaussian curvature of the Suita metric is bounded from above by $-4$, and moreover the curvature is identically equal to $-4$ if and only if $X$ is conformally equivalent to the unit disc less a (possible) polar set. After the contributions made by B\l{}ocki and Guan & Zhou, we provide an alternative and simplified proof of the equality part in Suita's conjecture. Our proof combines the Ohsawa-Takegoshi extension theorem and the plurisubharmonic variation properties of Bergman kernels.