Are all branch groups of intermediate growth just-infinite? I can't seem to find an answer to this one way or another; the question is motivated by the fact all examples of intermediate growth branch groups I know being just-infinite, and by this paper by Nekrashevych in which he constructs a family of branch groups, all but one of which are just-infinite and intermediate growth and one being neither.

This paper by Bartholdi, Grigorchuk and Sunic is my main reference on branch groups so far.

I've asked this question without success on math.SE two days ago, so I'm reposting it here hoping I'll be luckier.