Boyer–Gordon–Watson L-space/non-left-orderability conjecture

A nontrivial group $G$ is left-orderable if it admits a total ordering invariant under left multiplication. An $L$-space is a rational homology $3$-sphere with minimal Heegaard Floer homology, meaning that $\dim\widehat{HF}=|H_1(Y)|$. Boyer–Gordon–Watson conjecture. An irreducible rational homology $3$-sphere is an $L$-space if and only if its fundamental group is not left-orderable. This conjecture proposes an equivalence between Heegaard Floer-theoretic minimality and an order-theoretic property of the fundamental group; the source applies it to cyclic branched covers of pretzel knots, while its general status is not resolved here.