Reducibility reformulation of McDiarmid–Reed's conjecture

Let $G$ be a hexagonal graph, and let $\mathcal{F}_{9,4}$ and $\mathcal{F}'_{9,4}$ be the families of $(9,4)$-reducible H-handles used in the paper. Write $\operatorname{core}_{\mathcal{F}_{9,4}\cup\mathcal{F}'_{9,4}}(G)$ for the graph obtained after reducing $G$ by these configurations.

McDiarmid–Reed reducibility conjecture. For every hexagonal graph $G$, $\operatorname{core}_{\mathcal{F}_{9,4}\cup\mathcal{F}'_{9,4}}(G)$ is $(9,4)$-colorable.

The statement is presented as an equivalent form of McDiarmid–Reed's conjecture. The paper's new reductions substantially decrease the number of irreducible configurations found computationally, but some configurations remain and no proof is given.