Unsupervised speech models are becoming ubiquitous in the speech and machine learning communities. Upstream models are responsible for learning meaningful representations from raw audio. Later, these representations serve as input to downstream models to solve a number of tasks, such as keyword spotting or emotion recognition. As edge speech applications start to emerge, it is important to gauge how robust these cross-task representations are on edge devices with limited resources and different noise levels. To this end, in this study we evaluate the robustness of four different versions of HuBERT, namely: base, large, and extra-large versions, as well as a recent version termed Robust-HuBERT. Tests are conducted under different additive and convolutive noise conditions for three downstream tasks: keyword spotting, intent classification, and emotion recognition. Our results show that while larger models can provide some important robustness to environmental factors, they may not be applicable to edge applications. Smaller models, on the other hand, showed substantial accuracy drops in noisy conditions, especially in the presence of room reverberation. These findings suggest that cross-task speech representations are not yet ready for edge applications and innovations are still needed.