A stable shift-graph embedding conjecture

Let $T$ be a stable theory of graphs. For each cardinal $\mu$, let $\operatorname{ch_T}(\mu)$ denote the associated chromatic Hanf number, and let $\beth_{n-1}(\mu)$ be the corresponding beth iterate. Write $\text{Sh}_{m}(\omega)$ for the $m$-shift graph.

Stable shift-graph embedding conjecture. If, for every cardinal $\mu$,

$$\operatorname{ch_T}(\mu^+) \leq \beth_{n-1}(\mu)^+,$$

then for some $m\leq n$, every $\omega$-saturated model of $T$ contains an embedding of $\text{Sh}_{m}(\omega)$.

This asks for an analogue of the preceding characterization for simple theories and theories with stable edge relation, relating an upper bound on the chromatic Hanf number to the appearance of a finite shift graph in saturated models. The source presents it as a reasonable suggestion; no resolution is given.