Nonzero limiting value conjecture for the trapping-probability solution

Let $y(x)=y_j(x-x^*)$ for $x-x^*\in I_j$, where $y_j$ and $I_j$ are as in Theorem 4.1a, and let

$$L:=\lim_{x\to\infty}y(x).$$

Nonzero-limit conjecture. The limit $L$ exists and is different from zero.

This claim is posed to establish the required upper-boundary behavior for the trapping probability in the uniformly distributed proportional-loss model. The supplied text gives no resolution, so the conjecture remains open.