Do the finite subgraphs of one $\aleph_1$-chromatic graph realise every chromatic number? (Erdős #736)

Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that for every cardinal $m$ there exists a graph $G_m$ of chromatic number $m$ such that every finite subgraph of $G_m$ is isomorphic to a subgraph of $G$? In words: do the finite subgraphs of a single $\aleph_1$-chromatic graph already suffice to build graphs of arbitrarily large chromatic number? More generally, replace $\aleph_1$ by an arbitrary uncountable cardinal $\kappa$. (Erdős further asks to characterise the families $\mathcal{F}_\alpha$ of finite graphs for which some graph of chromatic number $\aleph_\alpha$ has all of its finite subgraphs in $\mathcal{F}_\alpha$.)

Acceptance. FULLY RESOLVES: a complete proof — machine-checkable (Lean/Coq) preferred, else fully written — either establishing Taylor's conjecture in ZFC (for every $\aleph_1$-chromatic $G$ and every cardinal $m$, a graph $G_m$ of chromatic number $m$ with all finite subgraphs embedding into $G$ exists), or refuting it in ZFC, or proving it independent by exhibiting a model where it holds to complement the Komjáth–Shelah model where it fails. ADVANCES (each checkable): improve the Komjáth–Shelah consistency result — they force a $G$ for which the finite-subgraph closure caps chromatic number at $\aleph_2$; stated in words, that cap is $\aleph_2$, so force a strictly smaller cap (e.g. $\aleph_1$) with proof; OR settle the conjecture under a specific axiom such as CH, GCH, or $V=L$; OR resolve the uncountable-$\kappa$ generalization for a specific $\kappa$; OR deliver a Lean/Coq formalization of the KoSh05 construction. Deliver the forcing/model construction or the proof object.