Maybe, but I would kind of assume that poorer people already end up emitting a lot less carbon since most carbon emitting activities already have a significant cost involved.
I'm not so sure: poorer people often have to drive further, in clunkier cars. In any event, it's likely more 'regressive' than income: a rich person might earn 10X, but generate 2X the carbon. I bet there's data for this, though.
Because of the refund, 2X the carbon doesn't mean 2X the taxes. Someone who emits 120% of the average amount of carbon pays twice as much as someone who emits 110% of the average, but emits less than 10% more.