Functorial construction of rigid dualizing DG modules

Let $A$ be a commutative DG ring in the situation of Definition 503, and suppose $(R_A,\rho_A)$ is a rigid dualizing DG module over $A$ relative to $\mathbb{K}$. Let $B$ be a DG ring and let $A\to B$ be cohomologically essentially finite type. If $A\to B$ is cohomologically pseudo-finite, define

$$R_B:=\operatorname{RHom}_A(B,R_A)\in\operatorname{\mathsf{D}}(B).$$

If $A\to B$ is cohomologically essentially smooth of relative dimension $n$, define

$$R_B:=\Omega^n_{B/A}[n]\otimes_A^{\mathrm L}R_A\in\operatorname{\mathsf{D}}(B).$$

Functorial rigidity conjecture. In the pseudo-finite case, $R_B$ has an induced rigidifying isomorphism $\rho_B$ relative to $\mathbb{K}$; in the essentially smooth case, $R_B$ likewise has an induced rigidifying isomorphism $\rho_B$ relative to $\mathbb{K}$.

These formulas propose the expected functorial behavior of rigid dualizing DG modules under the two specified classes of morphisms. The supplied text does not state that this conjecture has been proved in general.