Here we go, this picture is an HFT 1/0 labeled die with a 1/0 lug in it. Yeah, that fits....
I should mention that the lugs used in this demonstration were all starter lugs or the smallest OD lugs that will fit over 1/0, 2GA or 4GA wire.
The heavy duty lugs are considerably larger, OD wise. In fact a 1/0 heavy duty lug will not even fit between the jaws of this crimp tool when wide open!