Phantom category conjecture for Barlow surfaces

Let $X$ be a Barlow surface, and let $\mathrm{D}(X)$ denote its derived category. Barlow surface phantom conjecture. The category $\mathrm{D}(X)$ has an exceptional collection of length $11$ whose orthogonal complement is a non-trivial phantom category, and $\mathrm{D}(X)$ has no full exceptional collection. The source relates this conjecture to Kuznetsov's fullness conjecture and notes that a determinantal Barlow surface with an exceptional collection of line bundles of length $11$ and non-trivial phantom orthogonal complement is known; the nonexistence of a full exceptional collection remains the unresolved part.