Suppose that $Y$ is an integral homology $3$-sphere. Is it true that, for any $n \in \mathbb{N}$, there is always an irreducible representation $\pi_{1}(Y) \rightarrow SU(n)$?

I'm also happy to see the answer when the Lie group is replaced by $SL(n, \mathbb{C})$ or $PSL(n, \mathbb{C})$, but keeping irreducible unchanged.

Edit: In this paper, Zentner proved the statement positively for $SL(2, \mathbb{C})$. It might be true that there is a threshold $n$ such that there is no more $SU(N)$ irreducible representations for $N \geq n$. The question was stated in this way because constructing counter-examples to the vanishing statement might be easier.