Common finite-chromatic subgraph of two graphs of chromatic number $\aleph_1$ (Erdős #62)

If $G_1$ and $G_2$ are two graphs each of chromatic number $\aleph_1$ (the first uncountable cardinal), must there exist a graph $G$ of chromatic number $4$ — or even of chromatic number $\aleph_0$ — that is isomorphic to a subgraph of both $G_1$ and $G_2$? Here the chromatic number of an infinite graph is the least cardinal $\kappa$ for which the vertices can be properly coloured with $\kappa$ colours.

Acceptance. FULLY RESOLVES: a proof settling the question — either a proof that any two (equivalently, any finite family of) graphs of chromatic number $\aleph_1$ share a common subgraph of chromatic number $4$ (and/or $\aleph_0$), or a counterexample consisting of two graphs of chromatic number $\aleph_1$ whose common subgraphs all have chromatic number $\leq 3$, together with a proof (relative to a specified model of set theory, clearly flagged, if the answer turns out to be independent of ZFC). A machine-checkable proof is preferred, otherwise a full written proof. ADVANCES: (a) prove the $\aleph_0$ version while leaving the chromatic-number-$4$ version open, or vice versa; (b) prove Erdős's stronger conjecture that every $\aleph_1$-chromatic graph contains all $4$-chromatic graphs of sufficiently large girth (which implies the affirmative answer); (c) establish a consistency or independence result for either version. Deliver the proof file.