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Issue is most likely going to be a pressure switch or limit switch causing it to shut off.If it is a limit, the tech may be able to see something such as dirty coil,dirty filter or air flow problem that would make it cycle on high limit.If a pressure switch he may be able to notice an issue with vent or condensate drain blockage that could be causing the issue.
But your husband is right that ideally it would be best if you could get furnace to fail while he is onsite.
If you can watch the furnace and when it should be on but is not, take a look on the control board inside furnace and see if there is a blinking light to indicate what is failing. Then give me the model on furnace and tell me the flash pattern .If you have a multi meter I can guide you in troubleshooting at that time.
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