L-space conjecture of Boyer, Gordon and Watson

Let $M$ be a closed, connected, irreducible, orientable $3$-manifold. An L-space is a rational homology $3$-sphere whose Heegaard Floer homology has minimal possible rank. The fundamental group $\pi_1(M)$ is left-orderable when it admits a strict total order invariant under left multiplication.

Boyer–Gordon–Watson L-space conjecture. $M$ is not an L-space if and only if $\pi_1(M)$ is left-orderable.

This conjecture relates Heegaard Floer homology to an algebraic property of the fundamental group and is used in the paper to strengthen consequences for periodic L-space knots. Its general status is not resolved in the supplied text.