(In passing, triangle waves are spectrally identical to square waves, the only difference is the phase of the harmonics. This is said as a reminder that spectrum analyzers discard time and phase information, since triangle and square waves have very different behavior with respect to how fast the equipment has to slew in order to follow the rise-time of the square wave. Do square waves and triangle waves sound different, even though spectrally identical? I leave that as an exercise for the reader ... )
Anyway, returning to the topic at hand, we can exploit this harmonic structure of the square wave in order to cool the voice coil while we break in the driver. If we want to voice to really be moving, we select the FIFTH harmonic to be the same as the Fs of the driver - so, for example, if the free-air Fs is 60 Hz, then we tune the square-wave generator to 12 Hz. If we want the third harmonic, we tune the generator to 20 Hz. Simple.
I do feel the key concept is a waveform with extremely high acceleration. This will get the attention of the cone, and assist the chemical re-bonding and re-arrangement of the paper fibers in the cone. A square wave is nothing more than a repetitive series of step functions, which is exactly what we want to stimulate the break-in process.
Going along with that thought, hmm - maybe a test DVD-V recorded at 96/24 would be a good idea after all, since it could contain a series of test tones that gradually ramp up and down in level, maybe over a 60 dB range. Set the DVD player to "repeat" and leave the thing alone for a several hours.
But failing that, I would manually crank the generator up and down in level to let the thermal cycling do its work - a heat/cool cycle would probably be desirable. Maybe full power for five minutes, forty-five minutes at -40dB, and ten minutes at -20dB? (Full power, cool-down, and pre-heat cycle.)