Closed-form conjecture for the precoloring polynomial of the hexwheel network

Let $N$ be the network from the hexwheel example, let $m\geq 3$, and write $\operatorname{cp}_m(\lambda)=\operatorname{rcp}(M(N))$. Let $\omega_\pm=\lambda-2\pm\sqrt{\lambda^2+4}$. Hexwheel precoloring-polynomial conjecture. For $m\geq 3$,

$$\operatorname{cp}_m(\lambda+1)=\frac{\omega_+^m+\omega_-^m}{2^m}+(-1)^m(\lambda-m-1).$$

In particular, the polynomials satisfy

$$\operatorname{cp}_3(\lambda)=\lambda^3-6\lambda^2+14\lambda-13,$$ $$\operatorname{cp}_4(\lambda)=\lambda^4-8\lambda^3+28\lambda^2-51\lambda+41,$$

and

$$\operatorname{cp}_{m+2}(\lambda)=(\lambda-1)\operatorname{cp}_{m+1}(\lambda)+(\lambda+1)\operatorname{cp}_m(\lambda)+(-1)^m(-2\lambda+m-1).$$

The formula and recurrence were verified computationally for $m\leq 11$, but no proof or resolution is supplied in the source.