The amenability necessity conjecture for crossed-product transference

Let $\Gamma$ be a discrete group, let $\theta:G\to\operatorname{Aut}(\mathcal{M})$ be a trace-preserving action, and let $p\ne2$. Suppose there is a complete isometry

$$j_p:L_p(\mathcal{M}\rtimes_\theta\Gamma)\xrightarrow{\quad j_p\quad}\prod_{\alpha,\mathcal{U}}L_p(\mathcal{M}\,\overline{\otimes}\,\mathcal{B}(\ell_2\Gamma))$$

satisfying

$$j_p(\operatorname{Id}\rtimes T_m)=(\operatorname{Id}\otimes M_m)^{\alpha,\mathcal{U}}j_p.$$

Amenability necessity conjecture. Under these assumptions, the action $\theta$ is amenable. This conjecture would show that amenability is necessary for the transference results considered in the paper; the authors note that this necessity is not known and is suggested by related transference results.