Linear-growth Hilbert length conjecture for locally compact groups

Let $G$ be a locally compact, compactly generated group. A Hilbert length function with linear growth on $G$ is a Hilbert length function whose growth is linear. Linear-growth Hilbert length conjecture. If $G$ has a Hilbert length function with linear growth, then $G$ has a proper, cocompact action on a Euclidean space. In particular, if $G$ is discrete, then it must be virtually abelian. This conjecture asserts that linear Hilbert growth essentially forces a finite-dimensional Euclidean geometric action; its status is not resolved in the supplied text.