The spider's-web conjecture for the quite fast escaping set

Let $f$ be a transcendental entire function of order less than $1/2$. For each $\varepsilon>0$, define

$$\mu_{\varepsilon}(r)=M(r)^{\varepsilon},$$

where $M(r)$ is the maximum modulus, and let $Q(f)$ be the quite fast escaping set obtained from the iterates of $\mu_{\varepsilon}$. A spider's web is a connected set containing boundaries of a nested sequence of bounded simply connected domains whose union is the whole plane.

Quite fast escaping set conjecture. The set $Q(f)$ contains a spider's web. Consequently, $I(f)$ is a spider's web and is connected.

The conjecture extends results for several families of entire functions of order less than $1/2$. The authors state that they know no counterexamples, while the assertion remains open in general.