Speech super-resolution/Bandwidth Extension (BWE) can improve downstream tasks like Automatic Speaker Verification (ASV). We introduce a simple novel technique called Self-FiLM to inject self-supervision into existing BWE models via Feature-wise Linear Modulation. We hypothesize that such information captures domain/environment information, which can give zero-shot generalization. Self-FiLM Conditional GAN (CGAN) gives 18% relative improvement in Equal Error Rate and 8.5% in minimum Decision Cost Function using state-of-the-art ASV system on SRE21 test. We further by 1) deep feature loss from time-domain models and 2) re-training of data2vec 2.0 models on naturalistic wideband (VoxCeleb) and telephone data (SRE Superset etc.). Lastly, we integrate self-supervision with CycleGAN to present a completely unsupervised solution that matches the semi-supervised performance.