Orlov's rationality conjecture for surfaces with full exceptional collections

Let $X$ be a smooth projective surface. A full exceptional collection in $\mathrm{D}(X)$ is an exceptional collection that generates the derived category; in the line-bundle case, all its objects are line bundles. Orlov's rationality conjecture. If $X$ has a full exceptional collection, or a full exceptional collection of line bundles, then $X$ is rational. Smooth projective rational surfaces are known to admit full exceptional collections of line bundles by blow-up formulas and augmentation. The converse stated here remains an open problem.