We show that several leading pretrained facial recognition systems exhibit a variance dichotomy in their feature space. In other words, the feature vectors approximately lie in a lower dimensional linear subspace. We demonstrate that this variance dichotomy degrades the performance of an otherwise powerful scheme for anonymity/unlinkability and confusion attacks on facial recognition system devised by Zehavi et al. (2024), which is based on simple weight manipulations in only the last hidden layer. Lastly, we propose a method for the attacker to overcome this intrinsic defense of these pretrained facial recognition systems.