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The only thing I can offer is that 13.11 volts sounds okay because a "12v" battery typically reads 12.5-12.8 volts, and 13+ volts when it's being charged. I'm guessing you measured the voltage without disconnecting the battery and your rig is plugged into shore power and the onboard converter is charging the battery which can cause a 13.1 reading.
For example, disconnect the battery cables and the reading should drop to about 12.5 to 12.8 volts.
So the voltage looks good and your problem is at the furnace which is a 12v device on or off shore power.
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