DOF and focal reducers
 

I understand the mechanisms behind focal reducers, but I'm not sure about it's effects on depth of field. If it improves the light transmission by a stop, does it also give the equivalent DOF of a 1-stop faster lens? In other words, an f2.8 with a focal reducer might give me the transmissive equivalent of an f1.8, but does the depth remain the same as the f2.8 lens?

I'm assuming it would, since the reducer is simply shrinking (and concentrating) the image circle, not affecting the true aperature or any other factors that determine DOF.