a contradiction that $\psi$ is dangerous on both $C_1$ and $C_2$ and extends to a $3$-coloring $\varphi$ of $G$; then $\psi(v_2)=\psi(v_4)$, and because of the edge $v_2v'_2$, $\varphi$ is $v'_2$-diagonal on $C'_2$, and thus it is dangerous on $C'_2$. This is a contradiction by the induction hypothesis for $G'$. Therefore, if $\psi$ extends to a $3$-coloring of $G$, then it is dangerous on at most one of $C_1$ and $C_2$.

Suppose now that $\psi$ is dangerous on at most one of $C_1$ and $C_2$, and $n\ge 4$. We need to show that $\psi$ extends to a $3$-coloring of $G$. By a straightforward case analysis, this is true when $n=4$, and thus assume that $n\ge 5$. By symmetry, we can assume that $\psi$ is not dangerous on $C_2$, and thus $\psi(v_2)\neq \psi(v_4)$. Color $v'_2$ by $\psi(v_4)$ and give $v'_1$ and $v'_3$ distinct colors; the obtained coloring of $C'_2$ is $v'_1$-diagonal, and thus it is not dangerous on $C'_2$. By the induction hypothesis, we can extend the coloring to $G'$. This gives a $3$-coloring of $G$ extending $\psi$.

Therefore, the claim holds for reduced Thomas-Walls graphs. Suppose that $G$ is obtained from $G_0=T'_n$ by patching on an independent set $S$ and framing 