Non-isomorphic monadically equivalent order to the rational order

Let $R$ denote the rational order.

Non-isomorphic equivalent-order conjecture. There is an order $M$ such that $M$ and $R$ have the same monadic theory, but $M$ is not isomorphic to $R$.

The source explicitly marks this assertion as refuted by Gurevich. Thus no such order exists, according to the supplied status evidence.