Functoriality conjecture for wrong-way maps in KK-bordism theory

Let $f:X\to Y$ and $g:Y\to Z$ be two $K$-oriented smooth maps. Write $\Omega_!(g\circ f)\in \Omega_*(C^\infty_c(X),C_0(Z))$ for the wrong-way class, and let $(\mathpzc{E}_f,D_f)$ and $(\mathmathpzc{E}_g,D_g)$ be the cycles associated with $f$ and $g$. For a geometrically defined connection $\nabla$, consider their constructive unbounded Kasparov product.

Functoriality conjecture. The class $\Omega_!(g\circ f)$ should be identified, up to a canonical bordism, with

$$(\mathpzc{E}_f\otimes_{C_0(Y)}\mathpzc{E}_g,D_f\otimes 1_{\mathpzc{E}_g}+1\otimes_\nabla D_g),$$

and there should be a canonical bordism

$$(\mathpzc{E}_f\otimes_{C_0(Y)}\mathpzc{E}_g,D_f\otimes 1_{\mathpzc{E}_g}+1\otimes_\nabla D_g)\sim_{\rm bor}(\mathpzc{E}_{g\circ f},D_{g\circ f}).$$

The claim expresses functoriality of wrong-way maps in $KK$-bordism theory: composition of $K$-oriented smooth maps is represented by the constructive unbounded Kasparov product, up to a canonical bordism. The surrounding argument establishes the required bordism in the setting considered, but the scope of the general assertion beyond that construction is not specified in the supplied text.