Tritronquée asymptotics for vortex filaments at gradient catastrophe

Let $\beta(s,t,\xi)$ be the complex curvature-torsion variable for the vortex filament under the localized induction approximation, and let $(s_0,t_0)$ correspond to a point of gradient catastrophe. Let $a$ and $b$ be the constants appearing in the local Painlevé I reduction, let $\beta_0$ be the leading value, and let $\Omega_0$ denote the tritronquée solution of

$$\Omega_{\xi\xi}=6\Omega^2-\xi.$$

Vortex-filament tritronquée conjecture. The generic behavior near the gradient-catastrophe point is

$$\beta(s,t_0,\xi)\simeq\beta_0-\varepsilon\left(-\frac{36}{a^3b}\right)^{1/5}\Omega_0\left(\left(-\frac{a}{12b^3}\right)^{1/5}\frac{s-s_0}{\varepsilon^3}\right).$$

This is the vortex-filament analogue of the tritronquée behavior conjectured for generic NLS solutions near a critical point. The supplied source does not give evidence that this conjecture has been resolved.