Let $G$ be a group and let $c > 1$ be a constant. We denote by $B_c(G)$ the space of all coefficients of the representations of $G$ which are uniformly bounded by $c$; more precisely, a function $f: G \rightarrow \mathbb{C}$ belongs to $B_c(G)$ if there is a uniformly bounded representation $\pi: G \rightarrow \mathcal{B}(\mathcal{H}_\pi)$ such that $$ f(x) = \langle \pi(x) \xi, \eta\rangle $$ for some $\xi, \eta \in \mathcal{H}_\pi$.

Further, define $$ \| f \|_{B_c} := \inf \{ \|\xi\| \; \|\eta\| \} $$ where the infimum runs over all possibilities of $\pi$ as well as $\xi,\eta$. One can show that $B_c(G)$ is a Banach space.

Apparently it was conjectured for a while that $$ \bigcup_{c>1} B_c(G) = M_{cb}A(G) $$ where $M_{cb}A(G)$ is the space of completely bounded multipliers of the Fourier algebra of $G$ (which is isometrically isomorphic to the space of Herz-Schur multipliers of $G$). It is easy to show that $\bigcup_{c>1} B_c(G) \subseteq M_{cb}A(G)$ based on a complete characterization of the later space (by Bozejko-Fendler 91 and Jolissaint 92).

But Haagerup, in a never published manuscript 85, proved that this is not the case for $\Bbb{F}_N$, non-commutative free group with $N$ generators.

Question: Are there classes/examples of non-amenable groups for which the equation holds? What if we look at a locally compact group $G$ (instead of a discrete one)?