Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$ is an injective *-homomorphism from $\mathcal{A}$ into a subset of the bounded operators $\mathcal{B}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$, such that the resulting representation is irreducible. (A representation is irreducible if whenever $\pi(\mathcal{A})$ acts invariantly on a subspace $\mathcal{H}^\prime \subseteq \mathcal{H}$, then $\mathcal{H}^\prime = \mathbf{0}$ or $\mathcal{H}^\prime = \mathcal{H}$.)

Let $U \in \mathcal{B}(\mathcal{H})$ be any unitary ($U^* = U^{-1}$) bijection on $\mathcal{H}$, not necessarily in $\pi(\mathcal{A})$.

Does it follow that $U\pi(\mathcal{A})U^{*} = \pi(\mathcal{A})$?