A coefficient pattern conjecture for the characteristic polynomial of the substitution matrix

Let $n$ be odd with $n\geq 5$, let $M_n$ be the substitution matrix, and write its characteristic polynomial as

$$p_n(x)=-(x-1)q_n(x).$$

The coefficients of $q_n(x)$ in standard form are arranged according to the pattern described in the paper: their signs from left to right are $+,-,-,+,+,-,-,+,+,\ldots$, the last two coefficients in each row are $1$ or $-1$ depending on $n$, and each remaining coefficient is obtained by summing the absolute values of the two numbers indicated by adjacent segments of the same color and assigning the sign determined by its position. Coefficient-pattern conjecture. This coefficient pattern holds for all odd $n\geq 5$.