Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.

EDIT: As YCor rightly [pointed out](https://mathoverflow.net/questions/385829/are-locally-compact-hausdorff-locally-path-connected-topological-groups-locall#comment983279_385829), let's assume that $G$ is of finite topological dimension. The general theme of my question is to identify a minimal set of conditions on a topological group making it into a Lie group.