I have a (maybe dumb) question about the relation between the Künneth theorem and the Theorem-universal coefficient theorem (UCT for short) in $KK$-theory (for the setting see "The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor"- Rosenberg and Schochet).

Let $N$ be the bootstrap class of $C^*$-algebras (i.e. see here commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras) for the definition).

UCT: Let $A\in N$ and $B$ be $\sigma$-unital. Then there is a short exact sequence $$0\to Ext_{\mathbb{Z}}^1(K_*(A), K_*(B))\to KK_*(A,B)\to Hom(K_*(A), K_*(B))\to 0.$$

Künneth: Let $A$, $B$ be $\sigma$-unital and $K_*(B)$ finitely generated. Then there is a short exact sequence $$0\to K^*(A)\otimes K_*(B)\to KK_*(A,B)\to Tor_1(K^*(A),K_*(B))\to 0.$$

Is it possible to proof Künneth $\Rightarrow $ UCT, and if so, how to prove it (do you have a reference)? What is already known?