Poincaré functions with spiders' webs

For a polynomial $p$ with a repelling fixed point $z_0$, we consider Poincaré functions of $p$ at $z_0$, i.e. entire functions $\mathfrak{L}$ which satisfy $\mathfrak{L}(0)=z_0$ and $p(\mathfrak{L}(z))=\mathfrak{L}( p'(z_0) \cdot z)$ for all $z\in \mathbb{C}$. We show that if the component of the Julia set of $p$ that contains $z_0$ equals $\{z_0\}$, then the (fast) escaping set of $\mathfrak{L}$ is a spider's web; in particular, it is connected. More precisely, we classify all linearizers of polynomials with regard to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point $R$.