Maximizing-pattern conjecture for numerical ranges of cyclic shift matrices

Let $S(a_1,\dots,a_n)$ denote the $n\times n$ cyclic shift matrix with weights $a_1,\dots,a_n$. For $n\geq 2$, let $0\leq a_1\leq\cdots\leq a_n$ and set

$$A=S(a_1,a_3,a_5,\dots,a_6,a_4,a_2).$$

For a permutation $\sigma\in S_n$, let $A_\sigma$ be the cyclic shift matrix obtained by permuting the entries of $A$, and let $W(\cdot)$ denote numerical range. Maximizing-pattern conjecture. For every $\sigma\in S_n$,

$$W(A_\sigma)\subseteq W(A).$$

The stated pattern is known to maximize the numerical radius among permutations, and the conjecture asks whether it also maximizes the numerical range by set inclusion. The abstract reports this as established for $n\leq 6$ and conjectures that the pattern extends to general $n$; the general case remains open.