The phantom-category conjecture for Barlow surfaces and rational blow-downs

A phantom category is a nontrivial category with trivial $K^0$ group. Let $D^b(-)$ denote the bounded derived category of coherent sheaves.

Phantom-category conjecture. The derived categories of the Barlow surface and of the rational blow-down described in the source contain a phantom category as a semiorthogonal piece.

The conjecture places the phantom phenomenon in the semiorthogonal decompositions of these surfaces and is motivated by the associated compactified Landau–Ginzburg moduli spaces.