Correspondence homomorphism conjecture for Suslin homology

Let $\Bbbk$ be an algebraically closed field of characteristic $0$, and let $X,Y\in \mathbf{Sch}/\Bbbk$. For $\alpha\in c_{\mathrm{equi}}(X\times_{\Bbbk}Y/X,0)$, write $\mathrm{tr}_{\alpha}:H_i^S(X,\mathbb{Z})\rightarrow H_i^S(Y,\mathbb{Z})$ for the induced homomorphism. Here (M1) and (M2) are the properties defined earlier in the paper.

Correspondence homomorphism conjecture. (i) For any $\alpha\in c_{\mathrm{equi}}(X\times Y/X,0)$, $\mathrm{tr}_{\alpha}$ satisfies (M1). (ii) For any $\alpha\in c_{\mathrm{equi}}(X\times X/X,0)$, $\mathrm{tr}_{\alpha}$ satisfies (M2).

The conjecture generalizes the corresponding theorem from homomorphisms induced by morphisms to those induced by arbitrary correspondences. It has been confirmed by Bruno Kahn in the appendix.