4 problems

Infinite-family conjecture. If $T$ does not have the Schröder–Bernstein property, then there is an infinite collection of models of $T$ that are pairwise nonisomorphic and pairwise…

Naming-constants conjecture. (1) There is a small set $A\subseteq\mathfrak{C}$ such that $\operatorname{Th}_A(\mathfrak{C})$ has the Schröder–Bernstein property if and only if $T$…

Definable-set lifting conjecture. If $T'$ does not have the Schröder–Bernstein property, then $T$ does not have the Schröder–Bernstein property.

The classifiability and wandering-types conjecture. The complete theory $T$ has the Schröder–Bernstein property if and only if it is classifiable and has no wandering types.