Zhang's SU(2)-abelian rational homology sphere conjecture

Let $Y$ be a rational homology $3$-sphere. Call $Y$ $SU(2)$-abelian if every representation $\pi_1(Y)\to SU(2)$ has abelian image. Zhang's conjecture. If $Y$ is $SU(2)$-abelian, then $Y$ is a Heegaard Floer L-space. This conjecture predicts that the absence of irreducible $SU(2)$-representations forces the simplest possible Heegaard Floer homology. The paper proves the claim for the graph manifolds under consideration, but the general conjecture remains open.