Let $G$ be a bounded, connected, open set in $\mathbb{R}^2$, and let $\gamma_1$ and $\gamma_1$ denote two paths such that $\gamma_0(0)=\gamma_1(0)$, $\gamma_0(1)=\gamma_1(1)$, and the following hold for each $i=1,2$.
- For each $t\in[0,1)$, $\gamma_i(t)\in G$.
- $\gamma_i(1)\in\partial G$.
Say that a homotopy $\Gamma:[0,1]\times[0,1]\to\mathbb{R}^2$ from $\gamma_0$ to $\gamma_1$ is a $G$-homotopy if, for all $s\in[0,1)$ and all $t\in[0,1]$, $\Gamma(s,t)\in G$.
Let $w$ denote the final point of the $\gamma_i$ which is in $\partial G$. I believe that if $G^c\setminus\{w\}$ is contained in the unbounded face of $\gamma_0([0,1])\cup\gamma_1([0,1])$, then $\gamma_0$ and $\gamma_1$ are $G$-homotopic.
It seems clearly true, but it is not at all clear to me how to construct such a homotopy. Any help?
EDIT: I replaced "simply connected" with "connected" in the definition of $G$.