reference request – Unavoidable chords

A positive number $x$ is called an unavoidable chord, if it has this property:
For every continuous complex-valued function $gamma:(0,1)to C$ with $gamma(0)=0,gamma(1)=1$,
there exist $t_1neq t_2$ in $(0,1)$ such that $gamma(t_1)-gamma(t_2)=x$.

My colleague Mario Bonk proved
the following Theorem:
Unavoidable chords are exactly the numbers $1/n: n=1,2,3,ldots.$

He is curious whether this is a new result, and he sent his manuscript to several friends.
I am sure that I have seen this (or an equivalent statement) somewhere, don’t remember where, and my question is what is a reference.