It's obviously not impossible, just need to take it apart.

Or pry the door frame apart and tap it back in place after rolling the rack out.

Worth a mention as many door frames are easier to remove than a number of people might suspect .. fewer pieces to disassemble than many {object}'s and not an uncommon hack when moving furniture.

Removing door frames is easy. Putting it back together without looking like crap is the hard part.

Repeat after me: caulk and paint make it what it ain't

For the one I have that would be irreversible. You‘d have to cut it up.