(Non-)amenability of the Fourier algebra in the cb-multiplier norm

For a locally compact group $G$, let $A(G)$ denote its Fourier algebra, $M_{cb}(A(G))$ the completely bounded multipliers of $A(G)$, and $A_{M_{c}b}(G)$ the closure of $A(G)$ in $M_{cb}(A(G))$. We show that, if $A_{M_{c}b}(G)$ is amenable, then $a(G_d)$, the almost periodic compactification of the discretization of $G$, has an abelian subgroup of finite index. As a consequence, $A_{M_{c}b}(G)$ cannot be amenable if $G$ contains a copy of ${\text{free}}_2$, the free group in two generators, as a closed subgroup.