6 problems

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.…

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…

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

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 $R$.

Rational-order conjecture. There is a monadic sentence $\psi$ such that