The free-group conjecture on Fourier multipliers outside the Fourier–Stieltjes closure

Let $G=\mathbb{F}_n$ be the free group with $n{\geqslant} 2$. Let $\operatorname{VN}(\mathbb{F}_n)$ be its group von Neumann algebra, and let $M_\varphi$ denote the completely bounded Fourier multiplier associated with a multiplier function $\varphi$. Write $\overline{\operatorname{B}(G)}^{\mathrm{cb}}$ for the closure of the Fourier–Stieltjes algebra in the space of completely bounded Fourier multipliers.

Free-group Fourier multiplier conjecture. There exists a completely bounded Fourier multiplier

$$M_\varphi:\operatorname{VN}(\mathbb{F}_n)\to\operatorname{VN}(\mathbb{F}_n)$$

which does not belong to $\overline{\operatorname{B}(G)}^{\mathrm{cb}}$.

For discrete groups, decomposable Fourier multipliers coincide isometrically with the Fourier–Stieltjes algebra, so the conjecture would produce a completely bounded multiplier that is strongly non-decomposable. The source does not state a resolution.