Erdős problem 315

Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $\sum_{k\geq 1}\frac{1}{u_k+1}$ and $u_k=\lfloor c_0^{2^k}+1\rfloor$ for $k\geq 1$, where $$c_0=\lim u_n^{1/2^n}=1.264085\cdots.$$ Let $a_1<a_2<\cdots$ be any other sequence with $\sum \frac{1}{a_k}=1$. Is it true that $$\liminf a_n^{1/2^n}<c_0=1.264085\cdots?$$