This is in fact a simplified version of the question I've already posted. But I found it is interesting by its own.
Denote by $\Delta$ the unit disc in the complex plane and let $\gamma_1,\gamma_2: [0,1] \rightarrow \Delta$ be two paths in the unit disc such that $$\gamma_1(0)=\gamma_2(0)=0 $$ $$\gamma_1(1)=\gamma_2(1) \neq 0$$ $$\gamma_1((0,1)) \neq 0,\gamma_2((0,1))\neq 0$$
Prove (or disprove) that there exists a homotopy $H: [0,1]^2 \rightarrow \Delta$ between $\gamma_1,\gamma_2$ such that $$H((0,1) \times (0,1)) \neq 0$$
Notice that we can always find a homotopy between $\gamma_1,\gamma_2$ because $\Delta$ is simply connected. You can do the picture to move two path $[0,\frac{1}{2}]$ and $[0,\frac{1}{2}] * \partial \Delta(0,\frac{1}{2})$ ($*$ denotes the concatenate operation) which is passing through $0$ only at the boundary.
