Progress on Bondal–Orlov derived equivalence conjecture

In their 1995 paper, 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 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 any birational transformation between two $$3$$-dimensional Calabi–Yau varieties can be decomposed into a sequence of flops.

Ed Segal has also constructed an example in the $$n=5$$ case, and Daniel Halpern-Leistner 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 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?

• Kawamata proved the toric case. – Sasha Jun 19 '19 at 6:24
• @Sasha Thank you for this - I've added a link to the paper in my question. – mathphys Jun 19 '19 at 15:35