The conjectured inner amenability characterization by decomposable Fourier multipliers

Let $G$ be a locally compact group. The space of completely bounded Fourier multipliers on $\operatorname{VN}(G)$ is denoted by $\mathfrak{M}^{\infty,\mathrm{cb}}(G)$, and the space of decomposable Fourier multipliers by $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$. The Fourier–Stieltjes algebra of $G$ is $\operatorname{B}(G)$.

Inner amenability and decomposable multipliers conjecture. The group $G$ is inner amenable if and only if

$$\operatorname{B}(G)=\mathfrak{M}^{\infty,\mathrm{dec}}(G),$$

and the von Neumann algebra $\operatorname{VN}(G)$ is injective if and only if

$$\mathfrak{M}^{\infty,\mathrm{dec}}(G)=\mathfrak{M}^{\infty,\mathrm{cb}}(G).$$

These conjectures seek operator-algebraic characterizations of inner amenability and injectivity through decomposable Fourier multipliers. The source does not provide evidence resolving either assertion.