Uniqueness conjecture for the solution attaining the extremal minimum $\Xi_{\min}(x_0)$

Let $x_0\leq X(0)$, and let $\Xi_{\min}(x_0)$ be the extremal minimum location defined by

$$\Xi_{\min}(x_0)=\sup_{y_0,y_l}x_{\min}(x_0;y_0,y_l).$$

Uniqueness conjecture. For any $x_0$, the solution with a pole at $x_0$ and minimum at $\Xi_{\min}(x_0)$ is unique. The claim is presented as a numerical and heuristic extension of the preceding extremal-minimum uniqueness conjecture; no proof or resolution is supplied.