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AFAIK, a 6v generator may generate from time to time sufficient voltage (13.5 to 14.8v) to be passed on to the battery by a 12v regulator to charge a 12v battery, but not often enough to keep it charged properly. and so, the battery would slowly discharge.
...interesting question. why not give it a try and let us know how it goes? i'd be interested in hearing about it. i always find empirical evidence compelling.