On a conjecture of Erd\H{o}s and Graham about the Sylvester's sequence

Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $a_1 < a_2 < \cdots$ be any other positive integer sequence satisfying $\sum_{i=1}^\infty \frac{1}{a_i} = 1$. In this paper, we solve a conjecture of Erd\H{o}s and Graham, which asks whether $$\liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots.$$ We prove this conjecture using a constructive approach. Furthermore, assuming that the unproven claim of Erd\H{o}s and Graham that "all rationals have eventually greedy best Egyptian underapproximations" holds, we establish a generalization of this conjecture using a non-constructive approach. [This paper solves Problem 315 on Bloom's website "Erd\H{o}s problems".]