Uniqueness conjecture for Rumanov's beta-six auxiliary system

Let $(q_2(t),\alpha(t))$ denote solutions of the auxiliary nonlinear system associated with Rumanov's Lax pair, and impose the asymptotic conditions

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

Rumanov auxiliary-system uniqueness conjecture. The system has a unique smooth solution $(q_2,\alpha)$ satisfying these conditions as $t\to+\infty$. Such uniqueness would make the beta-six boundary-value construction canonical once the normalization of $\kappa$ and the branch of $u^{1/2}$ are fixed. The source states this as an additional conjecture and gives no resolution.