That answer does not make sense. If you have a typical AC audio signal, and you cut off the lower half of the waveform, say with a diode, you have a DC signal of no greater power. Plus, current is at zero for half of the time, therefore it would enjoy the same OR BETTER cooling time as the full AC.
The only real difference then is that the natural tension of the spider and surround are solely responsible for returning the speaker's cone to its original position. The lower half of the AC sinewave not being present to power the cone back.
This might work at lower frequencies, albeit not very efficiently, but certainly the physical structure could not respond in time for a high frequency signal. Plus, if the physical structure had enough strength to pull the cone back during the last half of a cycle, then it indoubtedly would have too much resistance to motion to operate efficiently during the first half, requiring MUCH more power to be applied for the same output.
There is one other issue: only a half-sinewave signal (AC with a diode) could be used with a speaker, not an actual DC signal. A "sinusoidal DC" signal IS an AC signal, just with the reference point moved. (A stepped DC could also be used because once agsin it is the essentially a rough sine wave or AC). Pure unmodified DC is a "square wave", which means current goes from zero to full power and back to zero in an instant, for each cycle. This creates enormous stresses compared to the gentle up and down motion of a sine wave (AC). DC literally will tear the cone apart at any good amount of power applied.
DC also does not sound "musical", so even if you could create a speaker cone that wouldn't be torn apart by reversing direction, it would not sound good.
So it appears that the issue is a more practical one of the physical, not an electrical one. The first answer above is false.