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

On the other hand, if there exists some $\varepsilon > 0$ such that $a_n$ satisfies $\liminf \frac{a_{n+1}}{a_n^{2+\varepsilon}} > 0$, then $a_n$ is an irrationality sequence by the above folklore result erdos_263.variants.folklore.