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Is there a simply connected smooth projective threefold $X$ with nef canonical divisor and vanishing third Betti number?

$X$ cannot have the same integral cohomology ring as the projective space (Fujita). $X$ cannot be of general type. The discussion in the introduction to this paper seems relevant.

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    $\begingroup$ Hirzebruch-Riemann-Roch shows $c_1c_2>0$ in this case so we cannot have Kodaira dimension $0$ either, as that forces the canonical class to be numerically trivial. $\endgroup$
    – Will Sawin
    Commented Apr 5, 2021 at 16:38

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Such a threefold does not exist, even assuming only $b_1(X)=0$ instead of $\pi_1(X)=0$. Indeed the Miyaoka-Yau inequality holds for projective manifolds with $K$ nef -- see this paper. For threefolds, this reads $K_X^3\leq -64\chi (\mathscr{O}_X)$, hence $\chi (\mathscr{O}_X)\leq 0$ because $K_X$ is nef. Since $h^1(\mathscr{O}_X)=0$, this implies $h^3(\mathscr{O}_X)\geq 1$, hence $b_3>0$.

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