The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure

In this paper, we introduce the concept of an A-by-FCE coarse fibration structure for metric spaces, which serves as a generalization of the A-by-CE structure for a sequence of group extensions proposed by Deng, Wang, and Yu. We show that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry that admit an A-by-FCE coarse fibration structure. As an application, the relative expanders constructed by Arzhantseva and Tessera, as well as the box space derived from an extension of Haagerup groups by amenable groups, are shown to exhibit the A-by-FCE coarse fibration structure. Consequently, their maximal coarse Baum-Connes conjectures are affirmed.