Kirby's $SU(2)$ representation conjecture for integer homology 3-spheres

Let $Y$ be a closed oriented 3-manifold satisfying

$$H_i(Y;\mathbb{Z})=H_i(S^3;\mathbb{Z}).$$

If $Y$ is not $S^3$, then Kirby's $SU(2)$ representation conjecture. there exists a homomorphism

$$\rho:\pi_1(Y)\to SU(2)$$

with non-abelian image.

This is the integer-homology-sphere case of a problem on whether every compact 3-manifold with nontrivial fundamental group admits an irreducible $SU(2)$ representation. Its significance comes from connections with instanton Floer homology, hyperbolic geometry, and quantum invariants; the conjecture remains open in the supplied source.