Sullivan's minimal Betti number conjecture for closed complex manifolds

Let $M$ be a closed complex manifold of real dimension $2n$, and let the total Betti number of $M$ mean the sum of all its Betti numbers. Sullivan's conjecture. The minimal total Betti number of a $2n$--dimensional closed complex manifold is

$$\min\{n+1,4\}.$$

The conjecture predicts that the topological complexity of closed complex manifolds initially grows linearly with dimension and then stabilizes. The paper confirms it in dimensions $2n\geq 8$, leaving dimension $6$ as the remaining case.