Eremenko's conjecture on unbounded components of the escaping set

Let $f$ be a transcendental entire function, and let

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

be its escaping set. Eremenko's conjecture. Every connected component of $I(f)$ is unbounded. The conjecture has been confirmed in a number of cases, but the general statement remains open; the source specifically notes results establishing it in several cases.