Bounded homomorphism characterization for Fourier algebras

Let $G$ and $H$ be locally compact groups. Suppose that $G$ is amenable, and let $\Phi:{\mathrm{A}(G)}\to{\mathrm{B}(H)}$ be a bounded homomorphism. A continuous map $\alpha:Y\subset G\to H$ is mixed piecewise affine when $Y$ partitions into finitely many elements of $\Omega(G)$, and on each piece the restriction agrees with either an affine or an anti-affine map as specified in the source. Bounded homomorphism characterization conjecture. Every such $\Phi$ is of the form $\Phi=\Phi_\alpha$ for some continuous mixed piecewise affine $\alpha:Y\subset G\to H$. The completely bounded analogue is established in the amenable case, whereas this bounded version is posed as an open question for Fourier algebras; the supplied text specifically describes characterizing bounded and completely bounded homomorphisms for non-amenable groups as open.