A structured numerical-range realization conjecture

Let $B\in \mathbb{C}^{d \times d}$, and suppose that its numerical range $\mathcal{W}(B)$ is invariant under multiplication by $e^{2\pi i/n}$. Let $\mathcal{C}_{\mathbb{C}}(n,d)$ denote the class of cyclic weighted shift matrices of size $d$ with cyclic structure parameter $n$. Structured numerical-range realization conjecture. There exists $A\in \mathcal{C}_{\mathbb{C}}(n,d)$ such that

$$\mathcal{W}(A)=\mathcal{W}(B).$$

If $\mathcal{W}(B)$ is also invariant under conjugation, the entries of $A$ can be chosen real. The preceding theorem establishes the analogous realization after enlarging the matrix size to $n\lceil d/n\rceil$; the conjecture asks whether the original size $d$ always suffices.