Vanishing matching-sum conjecture for odd square-grid cylinders

Let $C_{m,n}$ be the square-grid cylinder, let $Q_2$ be the class of configurations specified in the surrounding construction, and let $\pi$ be the associated projection map. For $\sigma\in Q_2$, consider all $\tau$ with $\pi(\tau)=\pi(\sigma)$. Vanishing matching-sum conjecture. If $n$ is odd, then

$$\sum_{\tau:\,\pi(\tau)=\pi(\sigma)}(-1)^{|\tau|}=0.$$

The supplied context gives no resolution and does not fully define $Q_2$ or $\pi$.