Also: the solubility of calcium carbonate goes down with increased temperature. Which is why your kettle gets covered in scale, while your toilet and sink generally don't.

Since the brushing process happens at a roughly constant temperature, this is going to behave more like a sink than any appliance that heats cold water and puts out hot.

I wouldn't worry about it if I were to try this. The fact that sponges turn gross in short order would be much higher up my list of reasons to be concerned.