Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}]_{1\le j,k\le n}$, where $$m_{jk}=\begin{cases}(1+\zeta^{j-k})/(1-\zeta^{j-k})&\text{if}\ j\not=k,\\1&\text{if}\ j=k.\end{cases}\tag{1}$$ Thus we have $$\det(M)=\prod_{s=1}^{n-1}(n+1-2s)=\begin{cases}(-1)^{n/2-1}\frac{((n-1)!!)^2}{n-1}&\text{if}\ 2\mid n,\\0&\text{if}\ 2\nmid n.\end{cases}.\tag{2}$$
What about the permanent of $M=[m_{jk}]_{1\le j,k\le n}$? I have the following conjecture based on my numerical computation.
CONJECTURE. If $n$ is a positive even number and $\zeta$ is a primitive $n$th root of unity, then for the matrix $M=[m_{jk}]_{1\le j,k\le n}$ with $m_{jk}$ given by $(1)$, we have $$\mathrm{per}(M)=((n-1)!!)^2.\tag{3}$$
QUESTION. Does the identity $(3)$ hold for any positive even number $n$?
Your comments are welcome!
PS: Let $n>1$ be odd. By arXiv:2206.02589, we have $$\det[m_{jk}]_{1\le j,k\le n-1}=(-1)^{(n+1)/2}\frac{((n-1)!!)^2}{n(n-1)}.\tag{4}$$ I also conjecture that $$\mathrm{per}[m_{jk}]_{1\le j,k\le n-1}=\frac{((n-1)!!)^2}n.\tag{5}$$
