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This letter studies the energy-constrained MMSE decentralized estimation problem with the best-linear-unbiased-estimator fusion rule, under the assumptions that 1. Each sensor can only send a quantized version of its raw measurement to the fusion center (FC), and 2. Exact knowledge of the sensor noise variance is unknown at the FC but only an associated statistical description is available. The problem setup relies on maximizing the reciprocal of the MSE averaged with respect to the prescribed noise variance distribution. While the considered design metric is shown to be highly nonlinear in the local sensor bit loads, we leverage several analytic approximation relations to derive an associated tractable lower bound; through maximizing this bound, a closed-form solution is then obtained. Our analytical results reveal that sensors with bad link quality are shut off to conserve energy, whereas the energy allocated to those active nodes is proportional to the individual channel gain. Simulation results are used to illustrate the performance of the proposed scheme.