Does the Kasparov product descend to KL-theory?

Determine whether the Kasparov product in KK-theory descends to KL-theory; specifically, ascertain if for all separable C*-algebras A, B, and C, the Kasparov product KK(A,B) × KK(B,C) -> KK(A,C) is compatible with the subgroup ZKK defining KL so that it induces a well-defined bilinear pairing KL(A,B) × KL(B,C) -> KL(A,C). Establishing this would allow arguments based on KK-equivalence to pass directly to KL-classes without additional assumptions.

Background

In Section 6.4 the authors compare their unified classification approach with the classical Kirchberg–Phillips argument. The classical proof yields that unitally KK-equivalent unital Kirchberg algebras are isomorphic. In their framework, conclusions at the level of KL-theory are essential, and passing from KK-equivalence to an isomorphism via KL would require compatibility of the Kasparov product with the KL quotient.

The authors explicitly note that their methods do not establish this compatibility, prompting them to obtain a substitute result (Proposition 6.21) under the additional assumption that there exists a *-homomorphism implementing the KK-equivalence. Under the UCT this extra assumption is unnecessary, but without it, whether the Kasparov product preserves KL remains an obstacle.