Slow-growing counterexamples to the strong Eremenko conjecture

Let $f\colon\mathbb{C}\to\mathbb{C}$ be a transcendental entire function. In 1989, Eremenko asked the following question concerning the set $I(f)$ of points that tend to infinity under iteration: can every point of $I(f)$ be joined to $\infty$ by a curve in $I(f)$? This is known as the \emph{strong Eremenko conjecture} and was disproved in 2011 by Rottenfu{\ss}er, R\"uckert, Rempe and Schleicher by the construction of a counterexample. The function has relatively small infinite order: it can be chosen such that $\log \log \,\lvert f(z)\rvert = (\log \lvert z\rvert)^{1+o(1)}$ as $f(z)\to \infty$. Moreover, $f$ belongs to the \emph{Eremenko--Lyubich class $\mathcal{B}$}. When a function belongs to this class, we can study the function via a \textit{logarithmic change of coordinates}. In this frame of coordinates, we are able to study the function via the \textit{tracts} that arise which are Jordan domains with unbounded real part. The key feature of the tracts in the counterexample of Rottenfu{\ss }er et al is that of large \textit{wiggling} sections. In this article we adapt the tracts used by Benitez and Rempe in order to deduce the existence of counterexample functions $f \in \mathcal{B}$ satisfying certain growth properties. We consider how slowly such an $f$ may grow. Suppose that $\Theta\colon [t_0,\infty)\to [0,\infty)$ is a function such that $\Theta(t) \to 0$ and $$(\log t)^{\beta \Theta(\log t)}\Theta(t) \to \infty \quad\text{ as $t\to \infty$}$$ for some $0<\beta<1$, along with a certain regularity assumption. Then there exists a counterexample $f\in\mathcal{B}$ as above such that $$\log \log \lvert f(z) \rvert = O\bigl( (\log \lvert z \rvert)^{1 + \Theta(\log \lvert z \rvert)}\bigr) \quad\text{as $f(z) \to\infty$}.$$ The hypotheses are satisfied, in particular, for $\Theta(t) = 1/(\log \log t)^{\alpha}$, for any $\alpha>0$.