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This paper proposes a distributed multiple constraints generalized sidelobe canceler (GSC) for speech enhancement in an N-node fully connected wireless acoustic sensor network (WASN) comprising M microphones. Our algorithm is designed to operate in reverberant environments with constrained speakers (including both desired and competing speakers). Rather than broadcasting M microphone signals, a significant communication bandwidth reduction is obtained by performing local beamforming at the nodes, and utilizing only transmission channels. Each node processes its own microphone signals together with the N + P transmitted signals. The GSC-form implementation, by separating the constraints and the minimization, enables the adaptation of the BF during speech-absent time segments, and relaxes the requirement of other distributed LCMV based algorithms to re-estimate the sources RTFs after each iteration. We provide a full convergence proof of the proposed structure to the centralized GSC-beamformer (BF). An extensive experimental study of both narrowband and (wideband) speech signals verifíes the theoretical analysis.