  We generalize Lusztig's geometric construction of the PBW bases of finite
quantum groups of type $\mathsf{ADE}$ under the framework of
[Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every
PBW basis of such quantum groups is proven to yield a semi-orthogonal
collection in the module category of the KLR-algebras. This enables us to prove
Lusztig's conjecture on the positivity of the canonical (lower global) bases in
terms of the (lower) PBW bases in the $\mathsf{ADE}$ case. In addition, we
verify Kashiwara's problem on the finiteness of the global dimensions of the
KLR-algebras of type $\mathsf{ADE}$.
