Universal uniqueness criterion?

[bo198214]

the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point.

Say the curve \( \gamma: [0,1]\to \mathbb{C} \) is injective and connects two points \( \gamma(0)=a \) and \( \gamma(1)=b \) with equal real part and with \( \Im(b)-\Im(a)>2\pi \). One needs to show that then there is always a pair of points \( c_1=\gamma(t_1) \) and \( c_2=\gamma(t_2) \) with equal real part and with \( \Im(c_2)-\Im(c_1)=2\pi \).

This sounds very plausible but I couldnt prove it except for certain simple shapes of \( \gamma \), e.g. convex.