5 problems

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

Consistency conjecture. The property $(*)$ is consistent with ZFC.

Boolean-algebra decidability conjecture. The theory of Boolean algebras of cardinality less than $\lambda$, or the first-order theory of Boolean algebras expanded by quantification…

Completion conjecture. There are orders with the same monadic theories whose completions do not have the same monadic theories.

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 $R$.