We present a convexly-constrained beamformer design for brain activity reconstruction from non-invasive electroencephalography (EEG) signals. An intrinsic gap between the output variance and the mean squared errors is highlighted that occurs due to the presence of interfering activities correlated with the desired activity. The key idea of the proposed beamformer is reducing this gap without amplifying the noise by imposing a quadratic constraint that bounds the total power of interference leakage together with the distortionless constraint. The proposed beamformer can be implemented efficiently by the multi-domain adaptive filtering algorithm. Numerical examples show the clear advantages of the proposed beamformer over the minimum-variance distortionless response (MVDR) and nulling beamformers.