Ramified étale–Brauer obstruction conjecture for stratified varieties

Let $k$ be a number field and let $X/k$ be a variety with $X(k)=\emptyset$. A stratification of $X$ is a decomposition

$$X=\coprod_i X_i.$$

For each stratum, let $\cl{X_i}$ be its closure and let $Y_i$ be a quasi-torsor over $\cl{X_i}$ that restricts to a torsor over $X_i$; for every twist $Y_i^{\sigma}$ of $Y_i$, let $Y_i^{\sigma}(\Ab_k)^{\Br}$ denote its Brauer set. Ramified étale–Brauer obstruction conjecture. There exists such a stratification and, for each $i$, such a quasi-torsor $Y_i$ satisfying

$$Y_i^{\sigma}(\Ab_k)^{\Br}=\emptyset$$

for all twists $Y_i^{\sigma}$ of $Y_i$. This conjecture would show that the absence of rational points is explained by the ramified étale–Brauer obstruction using one quasi-torsor for each stratum.