Existence of an ℓ2-invisible group of type F

Determine whether there exists a group Γ of type F (i.e., admitting a finite K(Γ,1)) that is ℓ2-invisible, meaning H_j(Γ; NΓ)=0 for all j (equivalently, H_j(Γ; ℓ2Γ)=0 for all j when Γ is of type F∞).

Background

The paper shows that ℓ2-invisibility (I∞) is bootstrappable and recalls constructions of ℓ2-invisible groups of type F∞ (e.g., certain local similarity groups). However, no example is known with a finite K(Γ,1), i.e., of type F.

Resolving this would impact the Zero-in-the-spectrum question and the landscape of ℓ2-invariants in geometric group theory.