Decidability of the monadic theory of orders of size at most $\aleph_1$

Consider linear orders of cardinality at most $\aleph_1$, and assume

$$\aleph_1<2^{\aleph_0}.$$

Their monadic theory is the collection of monadic sentences true in the orders under consideration.

Small-order decidability conjecture. The monadic theory of orders of cardinality at most $\aleph_1$ is decidable.

The source presents this as a conjecture after discussing decidability results for related structures. No resolution is supplied.