The radiation compatibility conjecture for multi-soliton singularity formation

Fix soliton parameters $A$, $\boldsymbol{\lambda}$, $\nu$, and $\sigma$. Let $\phi_{\mathrm{rad}}$ be a smooth solution of the wave equation $\Box\phi_{\mathrm{rad}}=0$ satisfying

$$\left\lVert \phi t^{\nu/2-1}\right\rVert_{L^\infty(\mathcal{M})}<\infty.$$

For each $z\in A$, write $t_z^{\nu/2-1}\phi_{\mathrm{rad}}|_{x_z=0}=\mathfrak{r}_z$. Radiation compatibility conjecture. There exists a smooth singularity-forming solution of the energy-critical wave equation satisfying the stated pointwise convergences if and only if there exists such a radiation solution for which

$$-\pi\nu\boldsymbol{\lambda}_0^{3/2}+\boldsymbol{\lambda}_0^2\sum_{\bar{z}\in A\setminus \{0\}}2\pi\sigma_{\bar{z}}\boldsymbol{\lambda}^{-1/2}_{\bar{z}}\frac{(1+\left\lvert \bar{z}\right\rvert)^{\nu/2}}{\left\lvert \bar{z}\right\rvert}\gamma_{\bar{z}}^{\nu/2-1}+2\pi\mathfrak{r}_0=0,$$

and the analogous Lorentz-boosted condition holds for every $z\in A$. The condition is proposed as a necessary compatibility requirement for constructing smooth multi-soliton singularities, but the source does not establish either implication.