Amenability characterization by completely bounded Fourier multipliers

Let $\Gamma$ be a discrete countable group. A Fourier multiplier of $\Gamma$ is completely bounded when it belongs to $M_0A(\Gamma)$, while $MA(\Gamma)$ denotes the Fourier multiplier algebra.

Amenability characterization. $\Gamma$ is amenable if and only if

$$M_0A(\Gamma)=MA(\Gamma).$$

Equivalently, every Fourier multiplier of $\Gamma$ is completely bounded exactly when $\Gamma$ is amenable. The result would characterize amenability through the boundedness properties of Fourier multipliers; the paper presents supporting examples, but the conjecture's resolution is not specified here.