

Increasing capacitance (how much charge is stored to reach a certain voltage) or the voltage it is charged to would indeed increase the capacity. Putting several in parallel would work, as would making a bigger capacitor. The main problem as far as I can tell is that the energy density of even supercapacitors is low, so you’d need a much larger volume to have the same capacity (and thus a much thicker phone).
The original research was regarding supercapacitors, not Lithium-ion. Based on the PNAS preview it seems it has to do with changing the design of a supercapacitor to increase charge rate.