Sup-norm closure of an ideal generated by an outer function in $H^\infty$

Question

Let $H^\infty$ be the space of all bounded analytic functions on the open unit disc $\mathbb{D}$ and let $f\in H^\infty$ be an outer function. Let $I$ be sup-norm closure of the principal ideal generated by $f$ in $H^\infty$.

My question is whether $I=H^\infty$.

Thanks in advance.