  In this paper we prove a conjecture stated by the first two authors
establishing the closure of the numerical range of a certain class of
$n+1$-periodic tridiagonal operators as the convex hull of the numerical ranges
of two tridiagonal $(n+1) \times (n+1)$ matrices. Furthermore, when $n+1$ is
odd, we show that the size of such matrices simplifies to $\frac{n}{2}+1$.
