Does Jesus Shave?

Tim Hughes wrote a song called ‘Jesus saves’. A few days ago it was in my head, but I was been singing the word ‘shaves’ instead of ‘saves’. I’m mature like that.
This got me thinking: does Jesus shave? On one level a silly, hypothetical question (probably on every level actually).

The growth of hair on humans is biologically a strategy for keeping warm – hair (or fur) acts as insulation, trapping a layer of air between hairs and preventing body heat from escaping.
Given that Jesus now has a resurrection body and is seated at the right hand of the Father in heaven, does he need this? Does his hair grow? Does he need (or choose) to shave?
This is a different question to the historical question of, as a first century Jew, was Jesus likely to have been bearded or not. It’s about the nature of the resurrection body, and, to an extent, heaven. If heaven is always warm (is it?), would body hair be necessary?

Any opinions? Does Jesus shave?