Boundary of Convergence of $\sum_{n=2}^\infty\frac{z^{2^n}}{\log(n)}$

Question

So yesterday I posted a question about the behavior of $\sum_{n=1}^\infty\frac{z^{2^n}}{n}$ on the unit circle and it was brought to my attention that for any $\epsilon>0,$ the sum $\sum_{n=1}^\infty\frac{z^{2^n}}{n^\epsilon}$ converges almost everywhere on the unit circle, despite there existing a dense subset of the unit circle where the sum is diverging to positive real infinity. So now I'm wondering if the same is true of $\sum_{n=2}^\infty\frac{z^{2^n}}{\log(n)}?$ Or, perhaps a more enriching question is, for the series $\sum_{n=1}^\infty\frac{z^{2^n}}{\phi(n)}$ where $\phi:\mathbb{N}\rightarrow(0,\infty)$ is monotonically increasing to infinity, how slow must $\phi(n)$ grow for this to stop being true?