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.