The L-space conjecture for closed orientable irreducible 3-manifolds

Let $M$ be a connected, closed, orientable, irreducible $3$-manifold. A left-orderable group is a group admitting a strict total order invariant under left multiplication, and a co-orientable taut foliation is a co-orientable taut foliation on $M$. The manifold $M$ is an L-space if it is a rational homology sphere with the minimal possible Heegaard Floer homology.

L-space conjecture. The following statements are equivalent for $M$: (1) $M$ is a non-L-space. (2) $\pi_1(M)$ is left orderable. (3) $M$ admits a co-orientable taut foliation $\mathcal{F}$.

The conjecture connects Heegaard Floer homology, left-orderable fundamental groups, and taut foliations. The implication from (3) to (1) is proved by Ozsváth and Szabó, so the equivalence is solved.