I just asked today the following question: On groups with finite pro-$p$ completion for all primes $p$. However, I can actually simplified what I am interested in, but as it is somewhat a more general question I will leave both of them. I would like to know of examples of finitely generated groups for which it is unknown whether their word growth is bounded below by $e^{\sqrt{n}}$. In particular, a result of Grigorchuk, see Theorem E.2 in https://arxiv.org/pdf/1512.07044.pdf, excludes many potential examples.