The vanishing conjecture for singularity-category K-theory of minimal models

Let $R$ be a 3-dimensional Gorenstein ring with rational singularities, and let

$$X\longrightarrow \operatorname{Spec} R$$

a minimal model. Write $\mathop{\rm D}_{\sf sg}(X)$ for the singularity category of $X$.

Vanishing conjecture. The Grothendieck group of the singularity category vanishes:

$$\mathop{\rm K}_{0}(\mathop{\rm D}_{\sf sg}(X))=0.$$

This conjecture asks whether minimal models in this setting have trivial singularity-category K-theory, even when the minimal model is not smooth and its singularity category is therefore generally nonzero. Its resolution status is not specified in the supplied text.