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

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