The converse Haagerup-property conjecture for locally compact wreath products

Let $B\leq A$ be locally compact groups, let $H$ act on a set $X$, and suppose that all point stabilizers are commensurated subgroups of $H$, with $A\neq B$. Write $B\wr^A_X H$ for the corresponding permutational restricted wreath product, and for a point stabilizer $L$ write $H\rightthreetimes L$ for the relative profinite completion.

Converse Haagerup-property conjecture. If $B\wr^A_X H$ has the Haagerup Property, then for every point stabilizer $L$ of $X$, the relative profinite completion $H\rightthreetimes L$ has the Haagerup Property.

This is proposed as the converse of the theorem giving sufficient conditions for the Haagerup Property of the wreath product. The source does not provide evidence of a resolution.