Taylor's conjecture on finite subgraphs of shift graphs

Let $G$ be a graph, and let $\chi(G)$ denote its chromatic number. For $n\in\mathbb{N}$, write $\operatorname{Sh}_n(\omega)$ for the corresponding shift graph on $\omega$. Taylor's conjecture. If

$$\chi(G)>\aleph_0,$$

then there exists an $n\in\mathbb{N}$ such that $G$ contains all finite subgraphs of $\operatorname{Sh}_n(\omega)$. The strong Taylor's conjecture was refuted, so this original conjecture is recorded here as a solved claim rather than as an open conjecture.