But unlike conventional batteries (where the supporting structure itself is lead and participates in the reaction, swelling, warping, crystalizing, flaking, or disolving) the graphite supporting structure's shape and bonding isn't affected at the voltages, temperatures, and chemistries involved.  So it remains intact.

The battery can probably be overvoltaged to drive off sulfation, or perhaps even to completely plate the lead over onto one plate or the other temporarily to break up even large third-stage crystals, without damaging the underlying structure.

The graphite structure, like activated charcoal, would have a HYSTERICALLY large surface area.  If you don't palte enough lead onto it to clog it, this would allow much higher operating currents and charge/discharge rates, too.