The tilting-bundle rationality conjecture for smooth projective surfaces

Let $X$ be a smooth projective surface. A tilting bundle on $X$ is a vector bundle $\cE$ such that $\cExt^i(\cE,\cE)=0$ for $i>0$, and the zero sheaf is the only sheaf $\cF$ satisfying $\cExt^i(\cE,\cF)=0$ for all $i$. Tilting-bundle rationality conjecture. If $X$ admits a tilting bundle, then $X$ is rational. Every rational surface admits a tilting bundle, but the converse is not known; the conjecture would characterize rationality among smooth projective surfaces by the existence of a tilting bundle.