As admitted on that page, there are numerous problems to be solved and I think that some may be insoluble. The mention of the problem that oxygen is causing in reacting with the electrolyte highlights a particular difficulty in that oxygen is a naturally excellent cathode in lithium battery chemistry. Thus there's two competing cathodes potentially in conflict, carbon and oxygen, the latter normally excluded in current (sic) cathodes.
Carbon isn't even considered an efficient cathode material and is excluded from the list of the thirteen most suitable elements, two of them in dual forms. The best is iron as used in the lithium iron phosphate (LiFePO4) batteries we have currently, so there has to be doubt that this proposed chemistry will ever become efficient enough.
Carbon as graphite is used as the anode in all existing lithium batteries where it's more efficient, but research is also being pursued to replace it in that application.