The $C$-numerical range and Unitary dilations

For an $n\times n$ complex matrix $C$, the $C$-numerical range of a bounded linear operator $T$ acting on a Hilbert space of dimension at least $n$ is the set of complex numbers $\mathrm{t}\mathrm{r}(C{X}^{\ast }TX)$, where $X$ is a partial isometry satisfying $X^{\ast }X=I_n$. It is shown that $$\mathbf{c}\mathbf{l}(W_C(T))=\cap \{\mathbf{c}\mathbf{l}(W_C(U)):U\text{ is a unitary dilation of }T\}$$ for any contraction $T$ if and only if $C$ is a rank one normal matrix.