Let $U,V$ be two unitary matrices in $\mathbb{U}(n)$. Let $f_U = A\mapsto UAU^*$ and $f_V=A\mapsto VAV^*$ be maps of type $\mathcal{M}_n\to \mathcal{M}_n$.
My question is whether $\left\lVert f_U-f_V\right\rVert_{op}\leq \left\lVert U-V\right\rVert_2$, where $\left\lVert-\right\rVert_{op}$ is the operator norm (where the domain and the codomain are equipped with the spectral norm) and $\left\lVert-\right\rVert_{2}$ is the spectral norm.
It is not hard to prove that $\left\lVert f_U-f_V\right\rVert_{op}\leq 2\left\lVert U-V\right\rVert_2$ but I am stuck to either find a counter-example or prove the former assertion.