Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $$\liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0$$ is not an irrationality sequence.

Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $$\liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0$$ is not an irrationality sequence.

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