Exact-to-nonexact spectrum comparison

Determine whether the definable spectrum of the simple exact nonnuclear algebra B=C(K)⋊rSL3(Z) is Borel-definably equivalent to the definable spectrum of the full group C*-algebra C*(F∞).

Background

The paper constructs B=C(K)⋊rSL3(Z) as a simple, exact, nonnuclear C*-algebra whose definable spectrum lies strictly above the common non-type-I nuclear degree. It also establishes comparisons involving the free group F∞ and explains that C*(F∞) is nonexact but not simple.

The unresolved question is the exact-to-nonexact comparison left open by the second half of Farah's Problem 9. The authors note that a simple-algebra version would additionally require a simple nonexact witness.