Let $\mathcal{A}$ be an abelian category, and $D$ its bounded derived category. An object $M \in D$ may be described as a list of cohomology objects $H^i = H^i(M)$ together with some complicated glueing data.

I am interested only in the case when $\mathcal{A}$ has homological dimension two. For example, $\mathcal{A}$ can be a category of coherent sheaves on a smooth surface. In this case the glueing data amounts to a collection of classes $\xi_i \in \mathrm{Ext}^2(H^i, H^{i-1})$ between each pair of adjacent cohomology objects, with no restrictions on the choices.

By definition, an object $M \in D$ is quasiisomorphic to a direct sum of complexes concentrated in a single degree (i.e., shifts of objects from $\mathcal{A}$) if and only if each $\xi_i$ vanishes.

Similarly, some objects in $D$ are quasiisomorphic to direct sums of complexes concentrated in two adjacent degrees. Is it possible to characterize this property by vanishing of some obstructions built in terms of the presentation of an object as a collection $\{ (H_i, \xi_i) \}_{i \in \mathbb{Z}}$ above?