Nonconvergent-scaling conjecture for approximate multi-solitons

Fix distinct soliton speeds, and let $f_{\mathrm{start}}$ denote the starting error whose leading behavior at one soliton is $t^{-1}$. Let $W_a$ be ground-state solitons, let $\tilde{\phi}$ be a correction, and write $\bar{\phi}=\sum_a W_a+\tilde{\phi}$. A nonconvergent-scaling conjecture. If $f_{\mathrm{start}}$ has leading $t^{-1}$ behavior at one of the solitons, then there is no approximate modulated multi-soliton solution with no outgoing radiation for which the soliton scaling is convergent. In particular, there exists $N>0$ such that no smooth function without outgoing radiation satisfying $|\tilde{\phi}|\lesssim t^{-1}$ obeys

$$|\Box\bar{\phi}+\bar{\phi}^5|\lesssim t^{-5}(t-r)^{-N}.$$

The claim is presented as a weaker version of the preceding nonexistence conjecture: the source explicitly clarifies that a genuine solution approaching a sum of solitons is not ruled out, but would have a nonconvergent scaling factor. No resolution is supplied in the text.