The growth of polynomials orthogonal on the unit circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$

We consider the polynomials $\{\varphi_n(z,w)\}$ orthogonal on the circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$ and show that $\|\varphi_n(e^{i\theta},w)\|_{L^\infty(\mathbb{T})}$ can grow in $n$ at a certain rate.
Bibliography: 21 titles.