  In this paper, we investigate the extremal values of (the logarithm of) the
characteristic polynomial of a random unitary matrix whose spectrum is
distributed according the Circular Beta Ensemble (C$\beta$E). More precisely,
if $X_n$ is this characteristic polynomial and $\mathbb{U}$ the unit circle, we
prove that: $$\sup_{z \in \mathbb{U} } \Re \log X_n(z) =
  \sqrt{\frac{2}{\beta}}
  \left(\log n - \frac{3}{4} \log \log n + \mathcal{O}(1) \right)\ ,$$ as well
as an analogous statement for the imaginary part. The notation $\mathcal{O}(1)$
means that the corresponding family of random variables, indexed by $n$, is
tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally
formulated for the case where $\beta$ equals to $2$, which corresponds to the
CUE field.
