Eremenko–Lyubich conjecture on detection of asymptotic values

Let $f\in\mathcal{S}$ be such that, for some $R>0$,

$$\liminf_{r\to\infty} \frac{1}{\log r} \int_{\{z\in\mathbb{C}\colon 1\leq |z|\leq r\text{ and }|f(z)|\leq R\}} \frac{\mathrm{d}x\,\mathrm{d}y}{|z|^2} > 0.$$

Eremenko–Lyubich's conjecture. Then $f$ has a finite asymptotic value. This conjecture connects the cylindrical area of large preimage sets with the existence of finite asymptotic values for finite-type entire functions; its resolution status is not specified in the supplied text.