Using a cat tree from a felv+ Cat

I rescued a very sick outside kiitty in september that turned out to be felv+. She has her own room and I had gotten her a cat tree but she has never used it at all, not even to scratch it. I would like to move it to a feral room, but want to be sure it would be safe to. i know the virus supposedly doesnt live long outside the cat's body so I had the cat tree out all week in freezing temperatures just to be sure. Any suggestions? Thanks!

According to Cornell University, the virus does not last long outside the cats body.  Not sure if cold would kill it but read this and decide.

http://www.vet.cornell.edu/fhc/Health_Information/brochure_felv.cfm

Thank you for caring for them all!

Thank you so much. That is a lot more reassuring.