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Conjecture 2 is already proved. Quote from J.A. Tilley, The a-graph coloring problem(2017): Theorem A.1. Let $G$ be an a-graph with boundary cycle $uxvy$ for the exterior 4-face and let $G$ have a 4-coloring $c$. Suppose, without loss of generality, that $c(x)=1$, $c(y)=1$ or 2, $c(u)=3$, and $c(v)=3$ or 4. Then there is either a 1–2 path between $x$ and $y$...

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