The spectral conjecture for hyperbolic groups

Let $\Gamma$ be a hyperbolic group. A uniformly bounded representation of $\Gamma$ is a representation on a Hilbert space whose operator norms are uniformly bounded. For every compact set $K\subset\Gamma$ and $\varepsilon>0$, seek a representation $\pi_{K,\varepsilon}$ on a Hilbert space $H_{K,\varepsilon}$, with no nonzero invariant vectors, and a unit vector $v_{K,\varepsilon}\in H_{K,\varepsilon}$ such that

$$\operatorname{diam}(\pi_{K,\varepsilon}(K)v_{K,\varepsilon})\leq\varepsilon.$$

There should also be a function $f:\Gamma\to\mathbb{R}_+$, independent of $K$ and $\varepsilon$, such that

$$\|\pi_{K,\varepsilon}(g)\|\leq f(g)\qquad(g\in\Gamma).$$

Spectral conjecture. There exists such a function $f$ and such representations and vectors for every compact $K\subset\Gamma$ and every $\varepsilon>0$. This would yield, by a limiting procedure, a representation with almost invariant vectors but no nonzero invariant vectors. The conjecture is motivated by the analogy between hyperbolic groups and rank-one lattices and by the role of complementary and principal series representations; its resolution status is not specified in the source.