As part of police inquiries into the blaze, a "number of names were looked at", said Christchurch Police detective inspector Tom Fitzgerald.
One of those suspects was Tully, the Herald understands.
However, the Winz killer was ruled out of being connected to the fire.
"There's nothing to link Mr Tully to that fire," Mr Fitzgerald told the Herald.
Nobody had been staying at the hut at the time of the fire, it's understood.
Pine Valley Hut was a comfortable 20-minute walk from Mill Flat camping ground and was a popular stop-off for children, families and Scout groups.
The police probe won't be reviewed "unless something new comes up", Mr Fitzgerald added.
A spokeswoman for DoC referred questions about the hut fire to police.
DOC has not built a replacement hut.
Tully will be sentenced for the Winz double-murder on April 15.