Existence of rigid dualizing DG modules for tractable commutative DG rings

Let $A$ be a tractable commutative DG ring, with traction $\mathbb{K} \to A$. A rigid dualizing DG module over $A$ relative to $\mathbb{K}$ is a pair $(R,\rho)$ consisting of a dualizing DG $A$-module $R$ and a rigidifying isomorphism $\rho:R\xrightarrow{\simeq}\operatorname{Sq}_{A/\mathbb{K}}(R)$ in $\operatorname{\mathsf{D}}(A)$. Existence conjecture. In the situation of Theorem 506, the rigid dualizing DG module over $A$ relative to $\mathbb{K}$ exists.

Existence is known when $A$ is an ordinary ring, but remains open for general tractable commutative DG rings; the more detailed conjectures in the paper are intended to imply this assertion.