The voltage changes in that case. In this case, a voltage difference exists between two materials with the same voltage relative to reference. It is as if you got pure gasoline out of a cars tailpipe.
Take the same capacitor, but make each plate into 2 layers. Pull the outer layers away and attach your electrodes to the inner layers. You can arrange to have more energy, at the same voltage. As the capacitor discharges, bring the outer layers in to keep the voltage across the inner plates constant. Reverse the process as it charges.
Please be clear, I'm not claiming that their battery works this way, or even that it works at all. I'm only making the claim that it is actually possible to design a device where what looks impossible here is not necessarily impossible.
From the paper:
> At the cathode, the build-up of plated metallic lithium changes the morphology of the cathode to increase C_C and create a very slow fade of V_cell with time of discharge