My purpose is to lower fs and lower tuning frequency which increased efficiency down low. Some modeling I did suggests that in an application I am looking at, lowering the fs of my driver from 25hz to 20hz and tuning appropriately, I would gain about 3db at 24hz which is equivalent to doubling the power needed to produce that frequency. It seems like a pretty good trade off. Reduce the total power the driver can take by reaching xmax with less power, but producing more of the frequency I want with less power.
Adding mass to the cone to lower Fs from 25 Hz to 20 Hz has a negligible (less than 1 dB) effect on efficiency south of 30 Hz, but it does significantly
decrease the efficiency higher up, which can make it
look like an increase in efficiency if you're comparing the -3 dB points. But from ballpark 30 hz on up, the higher Fs version will have higher efficiency.
Also, the suspension system (spider & surround) might not keep the voice coil centered in the gap over the years with the added mass, so you'd probably want to rotate the speaker 180 degrees every year or so.