Uniqueness conjecture for the positive dP_I^{(2)} solution

Let $\kappa,\tau\in\mathbb{R}$, and consider the dP$_{\rm I}^{(2)}$ equation for a sequence $\{\beta_n(\kappa,\tau)\}_{n\geq 0}$, with initial values specified by the source. A solution is called positive when $\beta_n(\kappa,\tau)>0$ for every $n\geq 1$. Uniqueness conjecture. For every $\kappa,\tau\in\mathbb{R}$, there is a unique positive solution of the dP$_{\rm I}^{(2)}$ equation satisfying

$$\beta_0(\kappa,\tau)=0,$$

and

$$\beta_n(\kappa,\tau)>0\quad\text{for all }n\geq 1,$$

corresponding to the stated initial values. This conjecture asserts that the orthogonality-motivated initial data select the unique globally positive solution, despite the sensitivity to perturbations and numerical precision described in the surrounding discussion. The source gives no resolution information.