[Kawamata showed][1] 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][2] 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.


  [1]: https://arxiv.org/pdf/math/0503102.pdf
  [2]: https://arxiv.org/pdf/math/0602258.pdf