Rigidity criterion for dualizing DG modules

Let $A$ be a tractable commutative DG ring with traction $\mathbb{K}\to A$, and let $(M,\rho)$ be a rigid DG module over $A$ relative to $\mathbb{K}$. Write $\bar A=\operatorname{H}^0(A)$ and let $\operatorname{Spec}\bar A$ be its spectrum. Assume that $M\in\operatorname{\mathsf{D}}^+_{\mathrm f}(A)$ and that $M$ is nonzero on each connected component of $\operatorname{Spec}\bar A$. Rigidity criterion conjecture. Then $M$ is a dualizing DG $A$-module.

This conjecture gives a criterion for recognizing dualizing DG modules from rigidity and finiteness. Its status is not resolved in the supplied text.