Parabolic-orbit conjecture for supercritical wave solitons

Let $\mathcal{D}^{\mathrm{par}}$ be a compactification of Minkowski space, let $W^{\mathrm{sup}}$ be the supercritical ground-state soliton, and let $z\in\mathbb{R}^3$. Define

$$y^{\pm}_{\mathrm{par}}=x\pm zt^{2/3}.$$

A parabolic-orbit conjecture. There exists a compactification $\mathcal{D}^{\mathrm{par}}$ and a polyhomogeneous function $\bar{\phi}$ on it solving the supercritical equation to $(t-r)^{-N}$ errors and satisfying

$$\lim_{t\to\infty}\left\lvert \bar{\phi}(x,t)-W^{\mathrm{sup}}(y^+_{\mathrm{par}})-W^{\mathrm{sup}}(y^-_{\mathrm{par}})\right\rvert_{L^\infty}=0.$$

Moreover, the value of $z$, up to rotational symmetry, is unique. This conjecture is motivated by the preceding construction of approximate supercritical multi-solitons and predicts a distinguished parabolic relative orbit; the supplied text gives no resolution.