Let $G$ be non-amenable finitely generated group.

1) Is it true that there exists a sequence $S(n)$ of sets which generate $G$ and such that

$\frac{1}{|S(n)|}||\sum_{g\in S(n)} \lambda(g)||\rightarrow 0$ when $n\rightarrow \infty$.

2) The same as (1), but $S(n)$ is finite subset of $G$.

Here $\lambda:G\rightarrow B(l^2(G))$ is left regular representation of $G$.

Also 1) is reformulation of 2).

**Edit:** here are some discussions on the question.