The definable-set lifting conjecture for the Schröder–Bernstein property

Let $T$ be a complete theory, let $M\models T$, and let $\varphi(\bar{x})$ be a formula of $T$. Let $T'$ be the theory of the structure $\varphi(M)$ equipped with all the definable structure induced from $M$.

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

The conjecture asks whether failure of the property on a definable set can always be lifted to failure for the whole theory. The paper explains that the naive lifting of bi-embeddable nonisomorphic models is technically difficult, and gives no resolution.