Limit conjecture for the stopping probabilities of a random walk

Let $p_n$ denote the sequence of stopping probabilities for the random walk considered in the paper. Limit conjecture.

$$\lim_{n\to\infty}p_n=\frac{1}{2}.$$

This is presented as the main question of the problem. The supplied text gives no resolution, so whether the stopping probabilities converge to one half remains open here.