Majority Dynamics conjecture for oscillating vertices in random regular graphs

Let $G$ be a random $d$-regular graph on $n$ vertices, and start Majority Dynamics from a random initial configuration. In a limiting cycle, a vertex is oscillating if its state changes between the two configurations of the cycle, rather than remaining fixed.

Majority Dynamics conjecture. For every $\varepsilon>0$, with high probability the number of oscillating vertices lies in

$$[(1/2-\varepsilon)n,(1/2+\varepsilon)n]$$

when $d$ is odd, and lies in

$$[0,\varepsilon n]$$

when $d$ is even.

This conjecture describes the experimentally observed parity-dependent behavior of Majority Dynamics on random regular graphs: approximately half the vertices oscillate for odd degree, whereas only a vanishing fraction oscillate for even degree. The statement is presented as a numerical conjecture, and no proof or resolution is given in the source.