I used liquid nitrogen on a 12" shaft weighing 400Kg. That was fun!
Carbide is probably only 4.5 parts per million per degree C
Steel is about 10.5
It will be a challenge to measure the change in size on your 6mm shaft.
With liquid nitrogen at -196 degrees it will shrink about 5 microns. -- 0.005mm -- about 2 tenths.
The fridge might give you 1 micron.
Super X3. 3600rpm. Sheridan 6"x24" Lathe + more. Three ways to fix things: The right way, the other way, and maybe your way, which is possibly a faster wrong way.