Derivatives at the zero point of a conformal map from the unit strip onto the unit disc

Question

Let $w$ be a point in the unit strip $\mathbb{S}=\{z\in \mathbb{C} : 0< Re\, z<1\}$ and let $\phi_w$ be a conformal map from $\mathbb{S}$ onto the unit disc $\mathbb{D}$ with $\phi_w(w)=0$. It is well-known that $\phi_w$ is unique up to multiplication by a fixed number of modulus $1$, and that $\phi'_w(w)\neq 0$.

Question: Is $\phi^{(k)}_w(w)\neq 0$ for every $k\in \mathbb{N}$?