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

Let $Y$ be a toroidal rational homology $3$-sphere. The refined toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 6$, then $Y$ is not $SU(2)$-abelian. This strengthens the preceding small-homology conjecture from homology order at most four to order at most six. The statement is known for toroidal graph manifolds when $|H_1(Y;\mathbb{Z})|\leq 5$, while the cases beyond that result remain open.