Binary Neural Networks (BNNs) are receiving an up-surge of attention for bringing power-hungry deep learning towards edge devices. The traditional wisdom in this space is to employ sign(.) for binarizing feature maps. We argue and illustrate that sign(.) is a uniqueness bottleneck, limiting information propagation throughout the network. To alleviate this, we propose to dispense sign(.), replacing it with a learnable activation binarizer (LAB), allowing the network to learn a fine-grained binarization kernel per layer - as opposed to global thresholding. LAB is a novel universal module that can seamlessly be integrated into existing architectures. To confirm this, we plug it into four seminal BNNs and show a considerable accuracy boost at the cost of tolerable increase in delay and complexity. Finally, we build an end-to-end BNN (coined as LAB-BNN) around LAB, and demonstrate that it achieves competitive performance on par with the state-of-the-art on ImageNet. Our code can be found in our repository: https://github.com/sfalkena/LAB.