"On the bit" in a bitless bridle?

My trainer started having me ride in a bitless bridle. I have a harder time rounding my horse in the bitless, and I'm just thinking about the term "on the bit" and whether it can apply to riding in a bitless bridle. Any thoughts on maintaining connection and roundness in a bitless bridle?

The bridle is like this:

If engagement and impulsion are coming from the hind end and the horse is using its abdominal muscles to lift its back, it will go "on the bit" in anything. Mine goes round in a flat halter with a lead rope clipped to each side. On the other hand, if the horse isn't engaged and has actually learned to put its head down in response to pressure from the bit, then you'll have more problems getting that if you go bitless.