Yau's almost-complex-to-complex conjecture

Let $n\geq 3$, and let $N^{2n}$ be a closed $2n$-dimensional manifold. An almost complex structure on $N$ is an endomorphism $J$ of the tangent bundle satisfying $J^2=-\operatorname{Id}$; a complex structure is an integrable almost complex structure.

Yau's conjecture. For $n\geq 3$, any closed $2n$-manifold admitting an almost complex structure will also admit a complex structure.

In real dimension six this includes the famous question of whether $S^6$ admits a complex structure. No closed $2n$-manifold with $n\geq 3$ is currently known to be almost complex but not complex, so the conjecture remains open.