ICAR-DMR finds compost properly pasteurised when it is held at 58 to 59 °C for 4 to 6 hours. It is not pushed there suddenly: the temperature climbs at about 1 °C per hour, reaching the target in 10 to 12 hours largely from the compost's own heat.
The ceiling is hard. Above 60 °C, ICAR-DMR says, all organisms are killed including the thermophilic fungi Phase II depends on, and the same manual names that as a cause of olive green mould. A tunnel needs a compost probe, not an air thermometer by the door.