Mine is on the wall,dead simple contact closure type when calling for heat, no connection to the air conditioner. The a/c has its own thermostat inside its interior cover.
Several other folks have posted that their integrated heat/cool thermostats have failed to turn on their furnaces lately due to loss of connection with the air conditioner, not sure which ones they have compared to yours.
If you want to confirm that the furnace works you can simply short the two wires from it to the tstat. Here is the suburban furnace manual
https://manuals.heartlandowners.org/manuals/Heating-Cooling/Furnace/Suburban/Suburban%20FURNACE_Service-and-Training-Manual-11-05-2015.pdf
It could be a problem with the furnace itself. If the furnace fan came on but no heat came out then look at the troubleshooting section. Could be the furnace never ignited.