Haagerup property conjecture for groups of local similarities of compact ultrametric spaces

Let $X$ be a compact ultrametric space, and let $LS(X)$ denote the locally compact group of local similarities on $X$. Haagerup property conjecture. The group $LS(X)$ has the Haagerup property. This would extend the known Haagerup-property results for groups arising from similarities of end spaces of trees to all compact ultrametric spaces; the status of the assertion is not resolved in the supplied text.