Symmetry conjecture for odd-sized modular sum-difference-closed subsets

Let $e$ be a positive integer, and let $S\subseteq\{0,1,\ldots,e-1\}$. Assume that for every $x,y\in S$, either $x+y\bmod e\in S$ or $x-y\bmod e\in S$. The odd-subset symmetry conjecture. Every odd-sized solution $S$ satisfies

$$0\in S,$$

and $S\cup\{e\}$ is symmetric about $e/2$. This gives necessary, but not sufficient, properties for the solutions of the associated open problem; the paper reports computational evidence for small values of $e$.