The categorical intersection conjecture for the varieties X_w

Let $Z$ be the variety introduced earlier in the construction, let $X_0$ be the variety associated with the zero face, and for each chamber indexed by $w$ let $X_w$ be the corresponding smooth variety. The maps from $Z$ to $X_0$ and to the $X_w$ induce pullback functors on derived categories. Categorical intersection conjecture. The intersection of the pullbacks of the categories $D^b(X_w)$ inside $D^b(X_0)$ is equivalent to $\operatorname{Perf}(Z)$. This conjecture is proposed as a categorical description of the analogue of the zeroth hypercohomology of the associated perverse schober; its status is not resolved in the supplied text.