Kawamata showed the derived category of coherent sheaves on a smooth projective toric variety has a full exceptional collection consisting of sheaves. I was wondering if it is know whether every smooth projective toric variety has a full exceptional collection consisting of vector bundles?
For context, King conjectured that a smooth complete toric variety has a full strong exceptional collection consisting of line bundles. This turns out to be false. For example, Hill and Perling constructed example of smooth projective toric variety which does not have a full strong exceptional collection consisting of line bundles. It seems significant work has gone into producing additional counterexamples, as well as, proving the conjecture under additional hypotheses.
I am wondering whether a similar conjecture might be true if we drop the strong hypothesis and allow vector bundles instead of line bundles.
Update: It was recently pointed out to me that Kawamata's paper contains a slight misstatement and the the full exceptional collection does not consist of just sheaves, but instead complexes of sheaves. See Remark 7 in the published version of "Derived categories of toric varieties II".