Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a modified homotopy equivalence, namely "partial homotopy in $X$" as following: Let $\gamma_1,\gamma_2: I \rightarrow \mathbb{C}^2$ ($I$ is the unit interval) such that $\gamma_1(0)=\gamma_2(0),\gamma_1(1)=\gamma_2(1)$. We say $\gamma_1$ and $\gamma_2$ are partially homotopic in $X$ if there exists a continuous map $H: I^2 \rightarrow \mathbb{C}^2$ such that $$H(\{0\} \times I)=\gamma_i(0),H(\{1\} \times I)=\gamma_i (1) $$ $$H(t,0)=\gamma_1(t),H(t,1)=\gamma_2(t) $$ $$H(Int(I^2)) \cap \Gamma = \emptyset $$ where $Int(I^2)$ is the interior of $I^2$, i.e. $(0,1)\times (0,1)$. We write $\gamma_1 \sim_X \gamma_2$ to indicate that $\gamma_1,\gamma_2$ are partially homotopic in $X$.

Now, let $\gamma \subset \Gamma$ be a path in $\Gamma$ and let $\gamma_1,\gamma_2$ be paths such that $$\gamma_i(0)=\gamma(0),\gamma_i(1)=\gamma(1) \, , i=1,2 $$ $$\gamma_i(Int(I)) \cap \Gamma =\emptyset $$ My question is: Suppose that $\gamma_1 \sim_X \gamma, \gamma_2 \sim_X \gamma$. Does it implies that $\gamma_1 \sim_X \gamma_2$?

My attempt: It is true if I take $\gamma$ such that $\gamma((0,1]) \subset X$ by combining the homotopy as usual. By same technique, I can build a homotopy $H$ between $\gamma_1$ and $\gamma_2$ such that $$H(Int(I^2)) \cap \Gamma =\gamma$$ It is very intuitively by myself that we can lift $H$ slightly to get away from $\Gamma$ but I am lacked of topology technique to do so and I don't know to to look at. Any advice is appreciate, even modification in the hypothesis, like restrict $H$ to be a embedding, etc. Thanks