Relative twist formula for étale characteristic classes

Let $S$ be a regular Noetherian scheme over $\mathbb Z[1/\ell]$ and let $f\colon X\to S$ be a smooth proper morphism purely of relative dimension $n$. Let $\mathcal F\in D_c^b(X,\Lambda)$ such that $f$ is universally locally acyclic relatively to $\mathcal F$. For any locally constant and constructible sheaf $\mathcal G$ of $\Lambda$-modules on $X$, let $\det\mathcal G(cc_{X/S}(\mathcal F))$ denote the rank-one locally constant and constructible sheaf on $S$ obtained from the reciprocity pairing. Relative twist formula. There exists a unique cycle class $cc_{X/S}(\mathcal F)\in {\rm CH}^n(X)$ such that

$$\det Rf_\ast(\mathcal F\otimes\mathcal G)\cong(\det Rf_\ast\mathcal F)^{\otimes\operatorname{rank}\mathcal G}\otimes \det\mathcal G(cc_{X/S}(\mathcal F))$$

in $K_0(S,\Lambda)$. This is a relative analogue of the twist formula for epsilon factors; the existence and uniqueness of the relative characteristic class are proposed as a conjecture.