Non-tangential ranges of holomorphic functions at Plessner points

Consider the random lacunary series $f(z) = \sum_{k=1}^\infty \frac{ξ_k}{\sqrt{k}} \, z^{2^k}$ on the unit disk, where $\{ ξ_k \}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $ζ\in \partial \mathbb{D}$ is a Plessner point of $f$, yet the image of ever