Toroidal graph manifolds with small homology are not SU(2)-abelian

We show that if $Y$ is a toroidal closed graph manifold rational homology $3$-sphere with $\mathrm{\mid }{H}_1(Y;\mathbb{Z})\mathrm{\mid }\le 5$, then there exists an irreducible representation $\text{fund}Y\to SU(2)$, using topological methods and avoiding the use of gauge theory. This answers positively to a conjecture by Baldwin and Sivek in the case of graph manifolds.