2004 Corvette battery dies

The battery dies frequently on my 2004 corvette. The dealership has done hours of monitoring and testing, has replaced the battery twice (the second time with a different model battery which seemed to do the trick for a while) but the problem continues to occur. When they fully charge the battery, we are able to go between 2 and 3 weeks without starting, and then when we attempt to start it, all is well. However, after we have started it and driven it for an hour or more, we can't let it sit idle for more than 4 or 5 days or it won't start (and the battery will be completely dead).

Anyone else had this experience?