Regularity of infinite exponentials

If a sequence $\{a_k\}_{k=0}^{\infty}$ is such that $a_k\ne 0$, $k=0,1,2,\dots$, and $\varlimsup_{n\to\infty}|a_n|=\bar a<\infty$, then
$$ f(z)=\lim_{n\to\infty}a_0z^{a_1z^{a_2z\cdots^{a_{n-1}z^{a_n}}}} $$
is regular in a domain $U$ such that $D\cap e^K\subset U$, where $D=\{z\colon|\arg z|<\pi\}$ and $e^K$ is the image of $K=\biggl\{w:|w|<\dfrac {1}{e\bar a}\biggr\}$ under the map $z=e^w$.