Bounded t-structure on the kernel MV_0(X, D) in higher dimensions

Determine whether, for semi-simplicial sets X of dimension greater than one with smooth G-action and any admissible collection D of subcategories, the pointwise t-structure on MV(X, D)^ω restricts to a bounded t-structure on the kernel MV_0(X, D)^ω (the kernel of α: MV(X, D) → D(G)), i.e., whether both truncation functors preserve MV_0(X, D)^ω and induce a bounded t-structure.

Background

The authors construct MV(X, D) as a compactly generated stable category of Mayer–Vietoris resolutions and show in rank-one cases that the kernel MV_0(X, D) inherits a bounded t-structure, enabling devissage and heart arguments for K-theory computations.

They note that when the building has dimension greater than one, it is not clear whether this bounded t-structure persists on MV_0(X, D)^ω. They explain implications (e.g., vanishing of K_{−1} and surjectivity on K_0 of the assembly map) and observe that recent results of Bartels–Lück remove certain obstructions, but the structural question about the t-structure restriction remains unresolved.