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Agreed, the battery is the most likely culprit. One that is been allowed to go almost completely dead will build up sulfation on the plates, and thus have a high internal resistance, even if it appears to take a charge. In addition, if it had any age on it before being allowed to sit, that will compound the problem.
The increased internal resistance means a sudden current drain, such as the fan switching on, will cause a more serious dip in voltage than would happen with a fresh battery possessing low internal resistance. At or near idle, the alternator won't be putting out enough power to make up the difference.
In practice, the ECU can function with quite a low voltage...but the fuel pump and injectors can't, and I expect the ignitor modules don't produce a lot of spark either. Result: engine stops.
Check connections at the battery terminals and the other connectors as Will suggests, but the odds are high that you will end up needing to replace the battery anyway. Have the charging system checked when you do that, just to be on the safe side. Good luck!
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John
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