The generalized flop conjecture for derived categories

Let $X$ and $X'$ be birational isomorphic varieties. Suppose that, for a resolution

\xymatrix{ & \widetilde{X}\ar[dl]_(.4){\pi} \ar[dr]^(.4){\pi'}&\\ X\ar@{-->}[rr]^{fl}&&X' }

of the birational isomorphism $X\stackrel{fl}{\dashrightarrow}X'$, one has $\pi^*K_X=\pi^{\prime*}K_{X'}$; such a birational transformation is called a generalized flop. Generalized flop conjecture. If $X$ and $X'$ are related by a generalized flop, then

$$\mathbf{D}^b(\operatorname{Coh} X')\cong\mathbf{D}^b(\operatorname{Coh} X).$$

The conjecture was proved for simple examples of flops, for arbitrary flops in dimension three, and for symplectic flops. The general statement remains open beyond these cases.