Monadic-theory characterization conjecture for orders in La08uchli's class

Let $\underline{M}$ be the class of orders generated from the one-point order by finite sums, products with $\omega$ and $\omega^*$, and the specified dense rational-indexed sums. For every $N\in\underline{M}$, there is a monadic sentence $\psi$ such that

$$M\models\psi$$

implies that $M$ and $N$ have the same monadic theory; it suffices to prove this for the rational order. Monadic-theory characterization conjecture. For every $N\in\underline{M}$ there is a monadic sentence $\psi$ such that $M\models\psi$ implies that $M$ and $N$ have the same monadic theory. The claim is presented as a question following the characterization of $\underline{M}$ by monadic sentences, and the supplied evidence indicates that it was refuted by Gurevich.