The infinite-family conjecture for failure of the Schröder–Bernstein property

Let $T$ be a complete first-order theory. Models are elementarily bi-embeddable when each elementarily embeds into the other.

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 elementarily bi-embeddable.

This asks whether one counterexample to the Schröder–Bernstein property can always be amplified to an infinite family of pairwise nonisomorphic, mutually elementarily bi-embeddable models. The paper presents it as an elementary-looking conjecture and does not resolve it.