The range of an arbitrary branch of $\log(z)$

This is a simple question, but it has been a while since I have studied complex analysis.

It is well known that on the complex plane with the nonpositive reals removed we can choose a branch of the complex logarithm whose range lies in the horizontal strip bounded by $\operatorname{Im}(z)=-\pi$ and $\operatorname{Im}(z)=\pi$. Such a branch is constructed explicitly in Ahlfors's book.

It is also well known that we can construct branches of the logarithm on any simply connected domain in $\mathbb C$ not containing zero.

My question: Does the range of a branch of $\log(z)$ formed from an arbitrary simply connected region necessarily lie in a bounded horizontal strip with width $2\pi$?

-

No. In fact, any curved branch cut (i.e. the domain is the complement of a smooth curve with no self-intersections starting at $0$ and extending out to $\infty$, not contained in a straight line) will give you a counterexample.