Mackey Borel structures of the CAR algebra and the crossed product

Determine whether the Mackey Borel structures of the CAR algebra and the crossed-product algebra B=C(K)⋊rSL3(Z) are isomorphic as measurable spaces.

Background

The paper distinguishes Borel-definable bijections, which require Borel lifts between representation codes, from isomorphisms of the associated quotient Mackey measurable spaces. It proves that the CAR algebra and B have different Borel-definable degrees, with the CAR degree strictly below that of B.

Question 6.1(i) asks whether this stronger Borel-liftable separation persists at the weaker level of bare Mackey Borel structures. The following discussion explains that any positive answer would necessarily be highly nonliftable: neither a Borel lift nor an m-measurable lift could exist.