SU(2) representation conjecture for closed 3-manifolds

Let $Y$ be a closed 3-manifold, and let $S^3$ denote the 3-sphere. A homomorphism from $\pi_1(Y)$ to $SU(2)$ is nontrivial if its image is not the identity subgroup.

SU(2) representation conjecture. If $Y\not\cong S^3$, then there is a nontrivial homomorphism

$$\pi_1(Y) \to SU(2).$$

This is a strengthening of the Poincaré Conjecture and addresses the existence of nontrivial $SU(2)$ representations of closed 3-manifold groups. The source presents it as an unanswered form of Kirby's Problem 3.105(A).