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In this article, we present characterizations of the concavity property ofminimal $L^2$ integrals degenerating to linearity in the case of fibrationsover products of open Riemann surfaces. As applications, we obtaincharacterizations of the holding of equality in optimal jets $L^2$ extensionproblem from fibers over products of analytic subsets to fibrations overproducts of open Riemann surfaces, which implies characterizations of theequality parts of Suita conjecture and extended Suita conjecture for fibrationsover products of open Riemann surfaces.

Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain VI: fibrations over products of open Riemann surfaces