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The success of DNNs is driven by the counter-intuitive ability of over-parameterized networks to generalize, even when they perfectly fit the training data. In practice, test error often continues to decrease with increasing over-parameterization, referred to as double descent. This allows practitioners to instantiate large models without having to worry about over-fitting. Despite its benefits, however, prior work has shown that over-parameterization can exacerbate bias against minority subgroups. Several fairness-constrained DNN training methods have been proposed to address this concern. Here, we critically examine MinDiff, a fairness-constrained training procedure implemented within TensorFlow's Responsible AI Toolkit, that aims to achieve Equality of Opportunity. We show that although MinDiff improves fairness for under-parameterized models, it is likely to be ineffective in the over-parameterized regime. This is because an overfit model with zero training loss is trivially group-wise fair on training data, creating an "illusion of fairness," thus turning off the MinDiff optimization (this will apply to any disparity-based measures which care about errors or accuracy. It won't apply to demographic parity). Within specified fairness constraints, under-parameterized MinDiff models can even have lower error compared to their over-parameterized counterparts (despite baseline over-parameterized models having lower error). We further show that MinDiff optimization is very sensitive to choice of batch size in the under-parameterized regime. Thus, fair model training using MinDiff requires time-consuming hyper-parameter searches. Finally, we suggest using previously proposed regularization techniques, viz. L2, early stopping and flooding in conjunction with MinDiff to train fair over-parameterized models.