Explicit description conjecture for the class of DG-flat modules

Let $R$ be the DG-algebra under consideration, and let ${\mathcal F}$ be the full subcategory of ${\mathsf{D}}(R)$ consisting of the relevant DG-flat objects. Let ${\mathcal F}' \subset {fmathsf{D}}(R)$ be the full subcategory of objects represented by DG-$R$-modules of the form $F^{0}\otimes_{R^{0}}R$ for some flat $R^{0}$-module $F^{0}$. It is immediate that ${\mathcal F}'\subseteq {\mathcal F}$. Explicit description conjecture.

$${\mathcal F}'={\mathcal F}.$$

This conjecture would give an explicit description of the class ${\mathcal F}$ in terms of flat modules over the degree-zero algebra $R^{0}$. The source presents this equality as still conjectural; no resolution is supplied.