The eigenvalue-form conjecture for the substitution matrix

Let $n$ be odd with $n\geq 5$, let $M_n$ be the substitution matrix, and let $s_{n,j}$ denote the edge lengths used in the construction, with $0\leq j\leq (n-1)/2$. The observed roots of the characteristic polynomial $p_n(x)$ are

$$\mu_n=\frac{1}{s_{n,(n-1)/2}},-\frac{1}{s_{n,(n-3)/2}},\ldots,\frac{1}{s_{n,2}},-\frac{1}{s_{n,1}},\frac{1}{s_{n,0}}=1.$$

Eigenvalue-form conjecture. Every eigenvalue of $M_n$ has the form $\pm\frac{1}{s_{n,j}}$ for all odd $n\geq 5$. If this holds and $g_n(x)$ is irreducible for odd $n\geq 11$, then $\mu_n$ is non-Pisot, because $\frac{1}{s_{n,(n-3)/2}}=\frac{1}{2\sin(3\pi/2n)}>1$.