Geometric rationality and affine stratifications for varieties with finite-dimensional DG models

Let $Z$ be a projective scheme, let ${\Bbbk}$ be the base field, and let $\mathscr R$ be a finitely-dimensional DG algebra. Suppose that the DG category $\mathscr{P}\!\mathit{erf}\!\operatorname{--} Z$ is quasi-equivalent to $\mathscr{P}\!\mathit{erf}\!\operatorname{--}\mathscr R$. Write $\widebar{Z}=Z\otimes_{{\Bbbk}}\widebar{{\Bbbk}}$. Geometric rationality and stratification conjecture. The scheme $Z$ is geometrically rational, and $\widebar{Z}$ has a stratification

$$\widebar{Z}=\bigcup_{i=1}^{k}Y_i$$

such that every $Y_i$ is an open subset of ${\mathbb A}^{n_i}$. This conjecture proposes a geometric description of projective schemes whose categories of perfect complexes admit finite-dimensional DG algebra models; the source says that even over an algebraically closed field of characteristic zero the problem is far from solved, and suggests that the asserted stratification may provide a complete description.