Finite-defect numerical-range conjecture for unitary dilations

Suppose $T\in {\mathcal L}({\mathcal H})$ is a contraction with

$$\dim{\mathcal D}_T=\dim{\mathcal D}_{T^*}=N<\infty.$$

Let ${\mathcal U}$ be the set of unitary $N$-dilations of $T$ to ${\mathcal H}\oplus \mathbb{C}^N$, and define $\tau:{\mathcal U}\to\mathfrak{K}$ by $\tau({\bf U})=\overline{W({\bf U})}$. Finite-defect numerical-range conjecture. The map $\tau$ wraps $\overline{W(T)}$. In particular,

$$\overline{W(T)}=\bigcap_{{\bf U}\in{\mathcal U}}\overline{W({\bf U})}.$$

Here $\mathfrak{K}$ is the ambient space of compact convex subsets in which the closures of the numerical ranges are viewed. The conjecture would complement Choi and Li's solved answer to Halmos's question, but the source states that it remains open even for $N=1$.