The SU(2)-abelian implies Heegaard Floer L-space conjecture

Let $Y$ be a rational homology $3$-sphere. The SU(2)-abelian implies Heegaard Floer L-space conjecture. If $Y$ is $SU(2)$-abelian, then it is a Heegaard Floer L-space. Here, a Heegaard Floer L-space is a rational homology $3$-sphere satisfying $\operatorname{rank} \widehat{HF}(Y)=|H_1(Y;\mathbb{Z})|$. This conjecture would connect the absence of non-abelian $SU(2)$-representations with the simplest possible Heegaard Floer homology. The source presents it as expected and uses it to motivate consequences for toroidal manifolds; no resolution is supplied.