Derived equivalence for the simple flop of type $G_2^{†}$ via tilting bundles

The aim of this article is to prove the derived equivalence for a local model of the simple flop of type $G_2^{†}$, which was found by Kanemitsu. This flop is the only known simple flop that comes from a non-homogeneous roof. The proof of the derived equivalence is done by using tilting bundles, and also produces a noncommutative crepant resolution of the singularity that is derived equivalent to both sides of the flop.