Conjecture on escaping sets in the Eremenko-Lyubich class

Let $\mathcal{B}$ denote the Eremenko-Lyubich class of transcendental entire functions whose set of singular values is bounded. For a transcendental entire function $f$, let

$$I(f):= \{ z \in \mathbb{C}: f^n(z) \rightarrow \infty \text{ as } n \rightarrow \infty \}$$

be its escaping set. A connected set $E\subset\mathbb{C}$ is a spider's web if there exists a sequence of bounded simply connected domains $(G_n)_{n\in\mathbb{N}}$ such that

$$\partial G_n \subset E, \quad G_n \subset G_{n+1}, \quad \bigcup_{n\in\mathbb{N}}G_n=\mathbb{C}.$$

Escaping-set conjecture. If $f\in\mathcal{B}$ is a transcendental entire function, then $I(f)$ is not a spider's web.

The paper proves this when $f$ has a finite logarithmic asymptotic value. The conjecture proposes that the same non-spider's-web conclusion holds for every transcendental entire function in the Eremenko-Lyubich class; its general status is not specified in the supplied text.