Let $(X\_t)$ be a non-negative supermartingale. Then for $s<t$,
$$ E[X\_t\mid \mathcal{F}\_s] \leq X\_s,$$
which can be written as
$$ -X\_s \leq E[-X\_t\mid \mathcal{F}\_s],$$
hence $(-X\_t)$ is a submartingale, and thus because the absolute value is convex, also $(|-X\_t|)$ is a submartingale. But because $X$ was non-negative, we have $|-X\_t| = X\_t$, so then we conclude that $(|-X\_t|) = (X\_t)$ is both a submartingale and a supermartingale, hence a martingale.

This is obviously wrong. But right now I don't see where I'm missing something.