Conjecture on the number of bi-embeddability classes of models of an age

Let $\mathfrak A$ be an age in a finite relational language, and let $\operatorname{Mod}(\mathfrak A)/{\equiv}$ denote the bi-embeddability classes of countable structures with age $\mathfrak A$. The bi-embeddability-class conjecture. The number $|\operatorname{Mod}(\mathfrak A)/{\equiv}|$ is finite if and only if

$$|\operatorname{Mod}(\mathfrak A)/{\equiv}|=1$$

if and only if $\mathfrak A$ is cellular. This would classify exactly when an age has finitely many bi-embeddability classes; the conjecture is presented as an open problem and is attributed to work cited as sib.