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Let X be a discrete metric space with bounded geometry. In this paper, we show that if X admits an "A-by-CE coarse fibration", then the canonical quotient map λ : C * max (X) → C * (X) from the maximal Roe algebra to the Roe algebra of X, and the canonical quotient map λ : C * u,max (X) → C * u (X) from the maximal uniform Roe algebra to the uniform Roe algebra of X, induce isomorphisms on Ktheory. A typical example of such a space arises from a sequence of group extensionsthat the sequence {N n } has Yu's property A, and the sequence {Q n } admits a coarse embedding into Hilbert space. This extends an early result of J. Špakula and R. Willett [25] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum-Connes conjecture holds for a large class of metric spaces which may not admit a fibred coarse embedding into Hilbert space.