The power-of-two Pfaffian conjecture for labeled Cauchon diagrams

Let $C$ be a labeled $m\times n$ Cauchon diagram, and let $\operatorname{Pfaffian}(C)$ denote its Pfaffian.

Power-of-two Pfaffian conjecture. The absolute value of the Pfaffian satisfies

$$|\operatorname{Pfaffian}(C)|=0$$

or is a power of $2$.

The conjecture is motivated by the observation that the Pfaffian of a $2\times n$ Cauchon diagram is always zero or plus or minus a power of $2$, and by analogous behavior observed for larger Cauchon diagrams. Its resolution is not indicated in the source.