Must every irrationality sequence $a_n$ in the above sense satisfy $a_n^{1/n} \to \infty$ as $n \to \infty$?

A folklore result states that any $a_n$ satisfying $\lim_{n \to \infty} a_n^{\frac{1}{2^n}} = \infty$ has $\sum \frac{1}{a_n}$ converging to an irrational number.