The toric fibration conjecture for full strongly exceptional collections

Let $X$ be a smooth complete toric variety. Suppose that $X$ is an $F$-fibration over $Y$, where $Y$ and $F$ are smooth toric varieties, and both $Y$ and $F$ have a full strongly exceptional collection of line bundles. Toric fibration conjecture. Then $X$ has a full strongly exceptional collection made up of line bundles. The conjecture extends constructions for several classes of toric fibrations and predicts that the existence of such collections on the base and fibre is inherited by the total space; the general case remains open.