Degree growth conjecture for homogeneous Gaussian cycle polynomials

For the $m$-cycle, let $\Gamma_m$ denote the homogeneous cycle polynomial defining the relevant algebraic boundary component. Homogeneous cycle-polynomial degree conjecture.

$$\deg(\Gamma_m)=m\cdot2^{m-3}.$$

The formula is predicted from the listed degrees for small $m$; the source explicitly states that an exact formula for the degree of the nonhomogeneous polynomial is unknown, and this homogeneous formula remains unproved.