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

Kovač and Tao [KoTa24] proved that any strictly increasing sequence $a_n$ such that $\sum \frac{1}{a_n}$ converges and $\lim \frac{a_{n+1}}{a_n^2} = 0$ is not an irrationality sequence in the above sense.

[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).