In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to determine the fibers of the Duflo map, and hence the left (or right) Kazhdan-Lusztig cells in the Weyl gorup for a Lie algebra of type $B$ or $C$.

Several places, both in reviews and in the papers themselves, mention is made of a fourth part which will deal with the additional things needed to treat type $D$, but I have not been able to find this paper anywhere (it is certainly not listed on MathSciNet).

What happened to this paper? Was it ever written? And if not, is there some other paper that describes what happens in type $D$?

shouldhave been posed as an answer (which I guess is now impossible, since your linked question is closed)? $\endgroup$