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For separable von Neumann algebras the Haagerup property for $N$ is equivalent to the existence of a real closable derivation $\delta$ such that $\delta^*\delta$ has compact resolvents in $\mathcal B(L^2(N))$. This is true even in the non-tracial case. See Theorem 7.7 in: Martijn Caspers, Adam Skalski, The Haagerup Approximation Property for von Neumann ...

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The answer is no. Let $G$ be a free abelian group of rank 2 generated by $x,y$. Let $S$ be the Meakin-Margolis expansion of $G$. It consists of all pairs $(X,g)$ with $X$ a finite connected subgraph of the Cayley graph of $G$ containing the origin and $g$. The product is $(X,g)(Y,h)=(X\cup gY,gh)$. The projection to $G$ is an idempotent pure homomorphism,...

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This is not an answer, just some sketchy thoughts that are too long for the comment box. I and some other HHS enthusiasts are very interested in this question being answered; we've tried a fair bit and have set it aside, so I don't think they'll mind me trying to recall what some of the strategies and issues are. It's indeed open for cocompact special ...

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Too long for a comment. Here is another approach when the free factors are torsion-free abelian groups. Let $G=\mathbb{Z}^{d_1}*\mathbb{Z}^{d_2}...*\mathbb{Z}^{d_n}$. You can realize $G$ as a generalized Schottky group, acting on the real hyperbolic space $H^n$ for some $n$ via a geometrically finite action. You first take a geometrically finite kleinian ...

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