The toroidal small-homology conjecture for SU(2)-abelian 3-manifolds

Let $Y$ be a toroidal rational homology $3$-sphere. The toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 4$, then $Y$ is not $SU(2)$-abelian. This conjecture predicts a non-abelian $SU(2)$-representation for every toroidal rational homology $3$-sphere whose first homology has order at most four. It is known for toroidal graph manifolds with $|H_1(Y;\mathbb{Z})|\leq 5$, but remains open in the stated generality.