Summary: We prove that the fundamental group of \(3\)-surgery on a nontrivial knot in \(S^3\) always admits an irreducible \(\operatorname{SU}(2)\)-representation. This answers a question of Kronheimer and Mrowka dating from their work on the property \(\mathrm{P}\) conjecture. An important ingredient in our proof is a relationship between instanton Floer homology and the symplectic Floer homology of genus-\(2\) surface diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to extend our main result to infinitely many surgery slopes in the interval \([3,5)\).