Barría–Halmos problem on the Cesàro operator

Let $H^2(\mathbb D)$ be the Hardy space of the unit disk. Define the Cesàro operator $C:H^2(\mathbb D)\to H^2(\mathbb D)$ by $C\left(\sum_{n=0}^{\infty}a_nz^n\right)=\sum_{n=0}^{\infty}\left(\frac{1}{n+1}\sum_{k=0}^{n}a_k\right)z^n$. For $\varphi\in L^{\infty}(\mathbb T)$, let $T_{\varphi}$ be the bounded Toeplitz operator on $H^2(\mathbb D)$, and let $\mathbf T=C^*(T_{\varphi}:\varphi\in L^{\infty}(\mathbb T))$ be the $C^*$-algebra generated by all bounded Toeplitz operators. Determine whether $C\in\mathbf T$.