Let $T\in\mathcal{B(\mathcal{H})}$ be a contraction and $X\in M_n$ with $\Vert X\Vert\leq 1$ s.t. $W(X)\subseteq \overline{W(T)}$ where $W(T)=\{\langle Tx,x\rangle :\Vert x\Vert=1\}$ is the numerical range of $T$. Can we conclude $X$ can be dilated to $T\oplus T\oplus\cdots\oplus T$?
Comments: I know this is true for the quadratic operator $T$ but in my case, $T$ is not necessarily quadratic. I have also observed that for a general $X$ with $W(X)\subseteq \overline{W(T)}$ (if we take out norm condition on $X$), it is not true. But I am clueless to conclude the above question.
Any hints/comments are highly appreciated. Thanks in advance.