An arbitrary-size cyclic weighted-shift conjecture for higher numerical ranges

Let $F_B\in\mathbb{C}[t,x,y]^{C_n}_d$ be the homogeneous polynomial associated with a matrix $B\in\mathbb{C}^{d\times d}$, and let $\mathcal{W}_k(B)$ denote its $k$-th numerical range. Let $\mathcal{C}_{\mathbb{C}}(n,d)$ denote the class of cyclic weighted shift matrices of size $d$ with cyclic structure parameter $n$. Higher numerical-range realization conjecture. There exists $A\in\mathcal{C}_{\mathbb{C}}(n,d)$ such that

$$\mathcal{W}_k(A)=\mathcal{W}_k(B)\qquad\text{for all }1\leq k\leq\left\lfloor d/2\right\rfloor+1.$$

The corollary proves this after replacing $d$ by $n\lceil d/n\rceil$; extending the theorem to arbitrary $d$ would imply the stated same-size realization, which remains unproved in the supplied text.