Open. The site labels the problem OPEN (page last edited 21 January 2026). Its commentary says that Kovač and Tao [KoTa24] have almost completely solved the problem by constructing a sequence growing doubly exponentially, $a_n^{1/\beta^n}\to\infty$ for some $\beta>1$, and that the remaining question is the exact exponent, in particular whether $\limsup a_n^{1/2^n}>1$ is possible, since a folklore result makes the sum irrational once $\lim a_n^{1/2^n}=\infty$. Their result is the accepted partial claim on the Kovač–Tao claim page (2024), every base $\beta<\sqrt{6/5}$. Two pending partial claims follow: a residual-state construction of 2026 asserts that every exponent $\beta<(\sqrt{13}-1)/2$ can be reached, and a Lean 4 development of 2026 asserts that $\limsup a_n^{1/2^n}>1$ is impossible. The problem asks for the exact growth threshold, which no claim determines.