Learned self-attention functions in state-of-the-art NLP models often correlate with human attention. We investigate whether self-attention in large-scale pre-trained language models is as predictive of human eye fixation patterns during task-reading as classical cognitive models of human attention. We compare attention functions across two task-specific reading datasets for sentiment analysis and relation extraction. We find the predictiveness of large-scale pre-trained self-attention for human attention depends on ‘what is in the tail’, e.g., the syntactic nature of rare contexts. Further, we observe that task-specific fine-tuning does not increase the correlation with human task-specific reading. Through an input reduction experiment we give complementary insights on the sparsity and fidelity trade-off, showing that lower-entropy attention vectors are more faithful.