I have a heat pump dryer, and I can confirm no shrinkage.
It's very gentle on clothes, but it does take a bit longer to dry.
Do you like it? I've toyed with the idea of getting one for energy efficiency, but my current washer and dryer (electric) are still chugging along and it feels wasteful
I absolutely shrunk my ex's marino wool in a heat pump dryer once. But I haven't had any issues with cotton.