Monotonicity conjecture for the even-indexed stopping probabilities

Let $p_n$ denote the sequence of stopping probabilities for the random walk considered in the paper. Even-index monotonicity conjecture.

$$p_{2n}\leq p_{2n+2}\quad\text{for all }ngeq 2.$$

The paper proves the corresponding inequality in the range treated by its preceding argument and observes the pattern in Table 1; the stated assertion extends this monotonicity to every $n\geq 2$.