I've used two different trainers with a variety of different tires over the last ~10 years. I didn't find a road tire that didn't spew black bits on the carpet over time. But the biggest problem is the treads always seems to glaze after a few sessions, so they slip and squeak during hard efforts.

My 1Up has a somewhat unrealistic power curve that is harder than regular road riding, so slippage can be common and annoying. I'm willing to be a guinea pig and try the Conti trainer tire. If, as the OP suggests, it slips less (or not at all, dare I hope?) then I'll be satisfied.