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Quantization is widely employed in both cloud and edge systems to reduce the memory occupation, latency, and energy consumption of deep neural networks. In particular, mixed-precision quantization, i.e., the use of different bit-widths for different portions of the network, has been shown to provide excellent efficiency gains with limited accuracy drops, especially with optimized bit-width assignments determined by automated Neural Architecture Search (NAS) tools. State-of-the-art mixed-precision works layer-wise, i.e., it uses different bit-widths for the weights and activations tensors of each network layer. In this work, we widen the search space, proposing a novel NAS that selects the bit-width of each weight tensor channel independently. This gives the tool the additional flexibility of assigning a higher precision only to the weights associated with the most informative features. Testing on the MLPerf Tiny benchmark suite, we obtain a rich collection of Pareto-optimal models in the accuracy vs model size and accuracy vs energy spaces. When deployed on the MPIC RISC-V edge processor, our networks reduce the memory and energy for inference by up to 63% and 27% respectively compared to a layer-wise approach, for the same accuracy.