In their 1995 [paper](https://arxiv.org/pdf/alg-geom/9506012.pdf), Bondal and Orlov posed the following conjecture:

> If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent sheaves are equivalent as triangulated categories, i.e. we have $D^b(\mathsf{Coh}(X)) \cong D^b(\mathsf{Coh}(Y))$.



**Selected 'particular' cases:**

[Tom Bridgeland](https://arxiv.org/pdf/math/0009053.pdf) proved that this holds in the $n=3$ case by showing that birational smooth projective Calabi-Yau threefolds are derived equivalent. This follows from the fact [that](https://projecteuclid.org/euclid.nmj/1118781185) any birational transformation between two $3$-dimensional Calabi–Yau varieties can be decomposed into a sequence of flops.

[Ed Segal](http://www.homepages.ucl.ac.uk/~ucaheps/papers/a%20new%205-fold%20flop%20and%20derived%20equivalence.pdf) has also constructed an example in the $n=5$ case, and [Daniel Halpern-Leistner](http://www.math.columbia.edu/~danhl/derived_equivalences_2016_09_18.pdf) has sketched a proof of the conjecture for the case of Calabi-Yau manifolds which are birationally equivalent to a moduli space of Gieseker semistable coherent sheaves (of some fixed primitive Mukai vector) on a K3 surface. As Sasha mentions in the comments, [Yujiro Kawamata](https://arxiv.org/pdf/math/0311139.pdf) has also proved the conjecture in the toric case.

There are also other cases in which the conjecture holds which I have not mentioned - thank you in advance for any comments/answers highlighting these.

**The general case?**

I have heard however that a proof of this conjecture in general seems rather far off at this moment in time. I am interested in whether any progress has been made with regards to the general case, and what approach/techniques may be involved in a potential proof?