The L-space conjecture for irreducible 3-manifolds

Let $M$ be a closed, connected, irreducible, orientable $3$-manifold. A Heegaard Floer $L$-space is a rational homology $3$-sphere with minimal Heegaard Floer homology; a co-orientable taut foliation is a taut foliation whose normal bundle is orientable; and a group is left-orderable if it admits a strict total order invariant under left multiplication.

L-space conjecture. The following statements are equivalent:

\text{$M$ is not a Heegaard Floer $L$-space$}, \text{$M$ admits a co-orientable taut foliation$}, $$\pi_1(M)\text{ is left-orderable}.$$

This conjecture seeks to relate Heegaard Floer homology, taut foliations, and orderability of fundamental groups. The paper proves it for irreducible $3$-manifolds admitting genus-one open book decompositions with connected binding, but it remains open in general.