Establishing certified uncertainty quantification (UQ) in imaging processing applications continues to pose a significant challenge. In particular, such a goal is crucial for accurate and reliable medical imaging if one aims for precise diagnostics and appropriate intervention. In the case of magnetic resonance imaging, one of the essential tools of modern medicine, enormous advancements in fast image acquisition were possible after the introduction of compressive sensing and, more recently, deep learning methods. Still, as of now, there is no UQ method that is both fully rigorous and scalable. This work takes a step towards closing this gap by proposing a total variation minimization-based method for pixel-wise sharp confidence intervals for undersampled MRI. We demonstrate that our method empirically achieves the predicted confidence levels. We expect that our approach will also have implications for other imaging modalities as well as deep learning applications in computer vision.