We study multi-task learning for two orthogonal speech technology tasks: speech and speaker recognition. We use wav2vec2 as a base architecture with two task-specific output heads. We experiment with different architectural decisions to mix speaker and speech information in the output sequence as well as different optimization strategies. Our multi-task learning networks can produce a shared speaker and speech embedding, which on first glance achieve a performance comparable to separate single-task models. However, we show that the multi-task networks have strongly degraded performance on out-of-distribution evaluation data compared to the single-task models. Code and model checkpoints are available at https://github.com/nikvaessen/disjoint-mtl