Finiteness conjecture for DNA origami monoids

Let ${\mathcal O}_n$ be the origami monoid of degree $n$ defined by the DNA-origami rewriting system. Finiteness conjecture. The order of ${\mathcal O}_n$ is finite for all $n$. This is motivated by computations for small values of $n$, but finiteness in general is not apparent from the definition and remains open.