Degree formula conjecture for Gaussian cycle varieties

Let $C_m$ be the $m$-cycle and let $P_{C_m}$ be the prime ideal of its Gaussian graphical model. Write $V(P_{C_m})$ for the associated projective variety. Degree formula conjecture.

$$\deg V(P_{C_m})=\frac{m+2}{4}\binom{2m}{m}-3\cdot2^{2m-3}.$$

The formula is conjectured from computations of the cycle varieties for small values of $m$ and remains open in general.