Stability under deformations of extremal almost-K\"ahler metrics in dimension 4

Given a path of almost-K\"ahler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-K\"ahler metric is an extremal K\"ahler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-K\"ahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.