Universal Painlevé-I2 scaling conjecture for generalized KP wave breaking

Consider the generalized KP equation

$$(u_t+u^n u_x+\epsilon^2u_{xxx})_x=\pm u_{yy},$$

with $\epsilon$-independent initial data $u(x,y,0;\epsilon)=u_0(x,y)$, and assume its solution is at least $C^4$ for $t>0$. Let $(x_c,y_c,t_c)$ be the critical point of the corresponding generalized dKP solution, let $u_c=u(x_c,y_c,t_c)$, and let $X$ and $T$ be the shifted variables defined in the source. Set $\kappa=-36G^c_{\xi\xi\xi}t_c^4$. Universal Painlevé-I2 scaling conjecture. In the limit $\epsilon\to0$, with $x\to x_c$, $y\to y_c$, and $t\to t_c$ so that $X/\epsilon^{6/7}$, $T/\epsilon^{4/7}$, and $(y-y_c)/\epsilon^{2/7}$ remain finite, the solution has the asymptotic expansion

$$u(x,y,t;\epsilon)=u_c+\frac{6}{nu_c^{n-1}}\left(\frac{\epsilon^2}{\kappa^2}\right)^{1/7}U\left(\frac{X}{(\kappa\epsilon^6)^{1/7}},\frac{T}{(\kappa^3\epsilon^4)^{1/7}}\right)+\bar y\left(F_y-F_\xi\frac{G_{\xi\xi y}}{G_{\xi\xi\xi}}\right)+O(\epsilon^{4/7}),$$

where $U$ is the specified particular solution of the second Painlevé-I equation. This conjectures a universal local profile at the first critical point, extending the one-dimensional KdV/Painlevé-I2 mechanism to generalized KP equations; its rigorous validity remains open.