Gasarch–Hirst graph-colouring conjecture

Let $G$ be a graph, let $\ell,k\in\mathbb{N}$, and let $\mathrm{COL}(\ell,k,G)$ denote the statement that if $G$ is locally $\ell$-colorable, then $G$ is globally $k$-colorable. Here $\mathrm{RCA}_0$ is the base theory of recursive comprehension and $\mathrm{WKL}$ is weak König's lemma.

Gasarch–Hirst conjecture.

$$\mathrm{RCA}_0\vdash (\forall \ell\geq 2)(\forall k\geq \ell)(\mathrm{WKL}\leftrightarrow\forall G\,\mathrm{COL}(\ell,k,G)).$$

The conjecture generalizes Gasarch and Hirst's theorem, which establishes the corresponding equivalence with $k=2\ell-1$. It asks whether every finite increase from the number of colors guaranteed locally to any $k\geq\ell$ yields the same reverse-mathematical strength as weak König's lemma.