The naming-constants conjecture for the Schröder–Bernstein property

Let $T$ be a complete theory in a monster model $\mathfrak{C}$. For a set $A\subseteq\mathfrak{C}$, write $\operatorname{Th}_A(\mathfrak{C})$ for the theory obtained by naming the elements of $A$.

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 $T$ is superstable and nonmultidimensional. (2) If $T$ has the Schröder–Bernstein property and $A\subseteq\mathfrak{C}$ is any set, then $\operatorname{Th}_A(\mathfrak{C})$ also has the Schröder–Bernstein property.

The conjecture concerns the effect of expanding the language by named constants. The paper notes that naming one constant can create the property, but conjectures that naming constants cannot destroy it; the second part would follow from the main characterization conjecture.