If $f$ is an non-null entire function that satisfies $|f(z)|=2$ for all $z\in\partial \Bbb D$, then $f$ is constant?

Question

I want to prove if either this statement is true or false:

Let $f:\Bbb C\to \Bbb C$ be a non-null entire function which verifies that $|f(z)|=2$ for every $z$ that belongs in the circle of center the origin and radius 1 ($\partial\Bbb D$, where $\Bbb D$ is the closed unit disc). Then $f$ is constant.

I tried starting with the maximum and minimum module princiles to prove it, but I got nowhere. I also tried looking for a counterexample, but I found nothing as well.

Could anyone please help me out?

EDIT: I understand that there is an answer here, but this is for bounded functions. Would it still work for non-bounded functions?