Rigidity of reducibility of Gevrey quasi-periodic cocycles on $U(n)$

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We consider the reducibility problem of cocycles (α,A) on T d × U(n) in Gevrey classes, where α is a Diophantine vector. We prove that, if a Gevrey cocycle is conjugated to a constant cocycle (α,C) by a suitable measurable conjugacy (0,B), then for almost all C it can be conjugated to (α,C) in the same Gevrey class, provided that A is sufficiently close to a constant. If B is continuous we obtain it is Gevrey smooth. We consider as well the global problem of reducibility in Gevrey classes when d = 1.