Language models (LMs) judges are widely used to evaluate the quality of LM outputs. Despite many advantages, LM judges display concerning biases that can impair their integrity in evaluations. One such bias is self-preference: LM judges preferring their own answers over those produced by other LMs or humans. The bias is hard to eliminate as frontier LM judges can distinguish their own outputs from those of others, even when the evaluation candidates are not labeled with their sources. In this paper, we investigate strategies to mitigate self-preference by reducing the LM judges' ability to recognize their own outputs. We apply black-box perturbations to evaluation candidates in pairwise comparison to obfuscate the authorship and reduce self-recognition. We find that perturbations as simple as synonym replacement for a few words predictably reduce self-preference. However, we also uncover fundamental challenges to eliminating the bias: when we extrapolate our perturbations to a more complete neutralization of stylistic differences between the evaluation candidates, self-preference recovers. Our findings suggest that self-recognition and self-preference can happen on many semantic levels, and complete mitigation remains challenging despite promising initial results.