Parity conjecture for the number of Sylow-2 cyclic factors

Let $M$ define a connected Cayley graph on $\mathbb{F}_2^r$, let $d(M)$ be the number of Sylow-$2$ cyclic factors in its sandpile group, and let the graph's eigenvalues be those associated with $M$. Parity conjecture. $d(M)$ is odd unless all eigenvalues have the same power of $2$; in that exceptional case,

$$d(M)=2^n-2.$$

This conjecture predicts a sharp parity dichotomy for the number of Sylow-$2$ factors, but the source provides no proof or resolution.