The numerical-range conjecture for periodic tridiagonal operators

Let $T=T(b,0,1)$ be the tridiagonal operator where $0$ and $1$ are the constant sequences of zeroes and ones, respectively, and let $b$ be the $(n+1)$-periodic sequence with period word $0^{n}1$. Let $J_n$ denote the $(n+1)\times(n+1)$ matrix with value $1$ at positions $(1,1)$ and $(n+1,n+1)$ and zero everywhere else, and let $B_n$ be the $(n+1)\times(n+1)$ matrix with $1$'s above the diagonal and zero everywhere else. For an operator $A$, write $W(A)$ for its numerical range, and let $\overline{W(T)}$ denote the closure of $W(T)$; $\operatorname{co}$ denotes the convex hull.

The numerical-range conjecture.

$$\overline{W(T)}=\operatorname{co}\bigl(W(B_n+J_n)\cup W(B_n-J_n)\bigr).$$

This conjecture describes the closure of the numerical range of this periodic tridiagonal operator in terms of the numerical ranges of two finite matrices. The displayed examples for $n=1,2,3$ provide supporting evidence, while the supplied text does not establish the equality or state its resolution.