Monotone subsequences conjecture for fractional-part sequences involving the Euler–Gompertz constant

Let $\delta$ denote the Euler–Gompertz constant, and let $p_k$ and $q_k$ be the sequences defined earlier in the paper. For the even and odd subsequences, consider

$$\delta\{q_{2k}\}-\{p_{2k}\}\quad (k=1,2,3,\ldots),$$

and

$$\delta\{q_{2k-1}\}-\{p_{2k-1}\}\quad (k=1,2,3,\ldots).$$

Monotone subsequences conjecture. There exists a subsequence of the first sequence that is monotone non-decreasing and a subsequence of the second sequence that is monotone non-increasing.

The claim is motivated by the observed distribution of these sequences, which appears close to uniform except near the endpoints; the source does not provide a resolution of the asserted subsequence property.