Rumanov's boundary-value conjecture for the beta-six Bloemendal–Virág equation

Let $u(t)$ be the Hastings–McLeod solution of the Painlevé II equation, let $(q_2(t),\alpha(t))$ solve the auxiliary system for Rumanov's Lax pair, and let $\kappa(t)$ and the branch of $u(t)^{1/2}$ be chosen in the gauge-transformed solution $\Psi(x,t)$. Define

$$F(x,t;\beta=6)=\Psi_{11}\bigl(3^{1/3}x,3^{2/3}t\bigr).$$

Rumanov's boundary-value conjecture. The function above determines the solution of the boundary-value problem for the Bloemendal–Virág equation if $(q_2,\alpha)$ satisfies

$$q_2(t)=-1+o(1),\qquad \alpha(t)=o(1),\qquad t\to+\infty,$$

and the normalization is fixed by

$$\kappa(t)u(t)^{1/2}\to1\qquad\text{as }t\to+\infty.$$

This identifies the beta-six distribution with the gauge-transformed Painlevé-II construction, subject to the stated boundary conditions. The paper presents it as Rumanov's conjecture and uses the associated nonlinear system to analyze the beta-six case; no resolution is supplied here.