The emergence of genomic language models (gLMs) offers an unsupervised approach to learn a wide diversity of cis-regulatory patterns in the non-coding genome without requiring labels of functional activity generated by wet-lab experiments. Previous evaluations have shown pre-trained gLMs can be leveraged to improve prediction performance across a broad range of regulatory genomics tasks, albeit using relatively simple benchmark datasets and baseline models. Since the gLMs in these studies were tested upon fine-tuning their weights for each downstream task, determining whether gLM representations embody a foundational understanding of cis-regulatory biology remains an open question. Here we evaluate the representational power of pre-trained gLMs to predict and interpret cell-type-specific functional genomics data that span DNA and RNA regulation. Our findings suggest that current gLMs do not offer substantial advantages over conventional machine learning approaches that use one-hot encoded sequences. This work highlights a major limitation with current gLMs, raising potential issues in conventional pre-training strategies for the non-coding genome.