Pearcey asymptotics conjecture for dissipative dKP near gradient catastrophe

Let $u(x,y,t;\epsilon)$ solve the dissipative dispersionless Kadomtsev–Petviashvili equation, and let $(x_c,y_c,t_c)$ be the first singularity of the corresponding dispersionless dKP solution. Define $X$ and $T$ as in the source's rescaling, and consider the double-scaling limit

$$\epsilon\to0,\qquad x\to x_c,\qquad y\to y_c,\qquad t\to t_c,$$

with $X/\epsilon^{3/4}$ and $T/\epsilon^{1/2}$ bounded. Let $u_c$ be the critical value, let $\bar y=y-y_c$, let $\bar\beta$ be the coefficient appearing in the local expansion, and set

$$\sigma=\epsilon\frac{6\left(1+c\left(t_cF_y^c\right)^2\right)}{F^c_{\xi\xi\xi}t_c^4}.$$

Pearcey asymptotics conjecture. Near the first singularity, the solution has the asymptotic expansion

$$u(x,y,t;\epsilon)\simeq u_c+\sigma^{1/4}U\left(\frac{X}{\sigma^{3/4}},\frac{T}{\sigma^{1/2}}\right)+\bar y\,\bar\beta+O(\epsilon^{1/2}),$$

where $U(a,b)$ is the Pearcey integral defined in the source.

This conjecture identifies the universal local profile of the dissipative dKP equation near gradient catastrophe with the Pearcey integral in the appropriate double-scaling regime. The supplied passage does not state whether the conjecture has been proved or disproved.