Let $H^2$ denote the Hardy space on the complex disk $D\subseteq \mathbb{C}$. Recall that for a function $f:D\rightarrow D$ the associated composition operator $C_f$ is defined by
$$
\begin{aligned}
C_f:H^2 &\rightarrow H^2\\
g &\mapsto g\circ f.
\end{aligned}
$$
We recall that $C_f$ is call cylic if there is some $g^f\in H^2$ such that $\operatorname{span}\{g^f\circ f^n\}_{n=0}^{\infty}$ is dense in $H^2$.
Lastly, let $L(H^2)$ denote the space of bounded linear operators on $H^2$ with the weak operator topology.
Is the set of composition operators $C_f$ for which $f$ is cylic, dense in $L(H^2)$? If yes, what about if it instead has the strong operator topology?