Stable-tree decomposition conjecture for finite models

Let $K$ be a class of finite models with a distinguished relation $\prec_K$, and let $P_1,\ldots,P_n$ be properties of $\langle K,\prec_K\rangle$. For a finite tree $T\subseteq{}^{\omega<}\omega$, let $\{M_\eta\prec_K M:\eta\in T\}$ be a family of substructures of $M$; call it a stable tree when it is stable in the sense of stable amalgamation.

Stable-tree decomposition conjecture. If $\langle K,\prec_K\rangle$ satisfies $P_1,\ldots,P_n$, then for every sufficiently large $M\in K$ there exist a finite tree $T\subseteq{}^{\omega<}\omega$ and a family $\{M_\eta\prec_K M:\eta\in T\}$ such that

$$\{M_\eta:\eta\in T\}\text{ is a stable tree},$$ $$\|M_\eta\|\leq n(K)\quad\text{for every }\eta\in T,$$

and $M$ is uniquely determined by $\bigcup_{\eta\in T}|M_\eta|$.

This is proposed as a decomposition theorem for sufficiently large finite models, analogous in spirit to structural decomposition results such as the theorem for finite abelian groups. The source does not establish the conjecture or specify the properties $P_1,\ldots,P_n$; it also notes that cardinality may not be the appropriate notion of smallness for the substructures.