Nonexistence conjecture for phantom categories on noncommutative projective planes

Let $A$ be a noncommutative projective-plane algebra, so that its noncommutative projective plane is represented by the category $\operatorname{qgr}(A)$. A phantom category is a nontrivial admissible full triangulated subcategory with trivial Grothendieck group and trivial Hochschild homology. Noncommutative-plane conjecture. Every noncommutative projective plane admits no phantom categories. The paper proves this for three-dimensional AS-regular quadratic algebras associated to nonsingular geometric triples $(E,\sigma,\mathcal{L})$ with $\sigma$ a translation of infinite order, but the argument does not extend to arbitrary noncommutative projective planes, so the general conjecture remains open.