Generalized accuracy conjecture for the near-field limit of high-order rogue waves

Let $\{K_n\subset\mathbb{R}^2\}_{n=1}^\infty$ be a sequence of compact sets satisfying

$$\lim_{n\to\infty}n^{-5/4}\sup_{(X,T)\in K_n}|X|=0\quad\text{and}\quad \lim_{n\to\infty}n^{-9/5}\sup_{(X,T)\in K_n}|T|=0.$$

Let $\psi_{2n}$ and $\psi_{2n-1}$ denote the even- and odd-order finite rogue waves, and let $\Psi^+$ and $\Psi^-$ denote their corresponding infinite-order near-field limits. Generalized accuracy conjecture. Uniformly for $(X,T)\in K_n$,

$$\lim_{n\to\infty}\sup_{(X,T)\in K_n}\left|n^{-1}\psi_{2n}(n^{-1}X,n^{-2}T)-\Psi^+(X,T)\right|=0$$

and

$$\lim_{n\to\infty}\sup_{(X,T)\in K_n}\left|n^{-1}\psi_{2n-1}(n^{-1}X,n^{-2}T)-\Psi^-(X,T)\right|=0.$$

This extends the near-field limit from fixed sets to sets whose spatial and temporal scales grow with $n$ under the stated bounds. In the original variables, $X=o(n^{5/4})$ corresponds to $x=o(n^{1/4})$, while $T=o(n^{9/5})$ corresponds to $t=o(n^{-1/5})$; the conjecture remains unverified in the supplied source.