What is the largest known second Betti number of a smooth projective threefold with nef canonical divisor and vanishing third Betti number?



Largest $b_2$ with nef $K_X$ and $b_3=0$
**Edit:** This example was wrong (it has $b\_3\neq 0$). In fact, such a threefold does not exist: this follows from my answer to [this question](https://mathoverflow.net/questions/389402/simply-connected-threefold-with-nef-k-x-and-b-3-0/389410#389410) (note that $b\_3= 0$ implies $b\_1= 0$ by Lefschetz theorem).


