Diffusion-based learning has settled as a rising paradigm in generative recommendation, outperforming traditional approaches built upon variational autoencoders and generative adversarial networks. Despite their effectiveness, concerns have been raised that diffusion models - widely adopted in other machine-learning domains - could potentially lead to unfair outcomes, since they are trained to recover data distributions that often encode inherent biases. Motivated by the related literature, and acknowledging the extensive discussion around bias and fairness aspects in recommendation, we propose, to the best of our knowledge, the first empirical study of fairness for DiffRec, chronologically the pioneer technique in diffusion-based recommendation. Our empirical study involves DiffRec and its variant L-DiffRec, tested against nine recommender systems on two benchmarking datasets to assess recommendation utility and fairness from both consumer and provider perspectives. Specifically, we first evaluate the utility and fairness dimensions separately and, then, within a multi-criteria setting to investigate whether, and to what extent, these approaches can achieve a trade-off between the two. While showing worrying trends in alignment with the more general machine-learning literature on diffusion models, our results also indicate promising directions to address the unfairness issue in future work. The source code is available at https://github.com/danielemalitesta/FairDiffRec.