Relative push-forward conjecture for characteristic classes

Let $S$ be a smooth connected scheme over a perfect field $k$ of characteristic $p$. Let $f\colon X\to S$ and $g\colon Y\to S$ be smooth morphisms of relative dimensions $n$ and $m$, respectively. Let $D_c^b(X/S,\Lambda)$ be the thick subcategory of objects $\mathcal F\in D_c^b(X,\Lambda)$ for which $f$ is $SS(\mathcal F,X/k)$-transversal, and define $K_0(X/S,\Lambda)$ as its Grothendieck group; define $D_c^b(Y/S,\Lambda)$ and $K_0(Y/S,\Lambda)$ analogously. For a proper morphism $h\colon X\to Y$ over $S$, relative push-forward conjecture. The diagram

$$\begin{CD} K_0(X/S,\Lambda) @>{cc_X^n}>> CH^n(X)\\ @V{h_\ast}VV @VV{h_*}V\\ K_0(Y/S,\Lambda) @>{cc_Y^m}>> CH^m(Y) \end{CD}$$

commutes; equivalently, for every $\mathcal F\in D_c^b(X,\Lambda)$ such that $f$ is $SS(\mathcal F,X/k)$-transversal, $h_\ast(cc_X^n(\mathcal F))=cc_Y^m(Rh_\ast\mathcal F)$ in $CH^m(Y)$. This extends the known degree-zero compatibility over finite fields; its validity in the stated relative setting is open.