Conjecture on the position of lumps emerging from the soliton front

Let $\epsilon$ be the small dispersion parameter, and let $a$, $b$, and $c_0$ be constants. Let $(x_0,y_0)$ be the position where a singularity of the Whitham system is expected to appear at time $t_0$, and let $(x_p,y_p)$ be the position where a lump is expected to appear at time $t_p$. Lump-position conjecture. The position of the lumps emerging from the soliton front is determined by

$$z_p=\dfrac{c_0}{\epsilon^{\frac{4}{5}}}\left[x_p-(a^2+3b^2)t_p-(x_0-(a^2+3b^2)t_0)+(a+\operatorname{i}\sqrt{3}b)(y_p+2at-y_0-2at_0)\right].$$

This conjecture connects the universal scaling near a Whitham-system singularity with the location of the KPI lumps emerging from the soliton front. The source presents it as a conjecture based on the connection between the NLS breather and KPI lump solutions; its resolution is not established in the supplied text.