Godbole et al.'s eventual monotonicity conjecture for ordered set partitions

Let $p$ be a pattern, let $k$ be an integer, and let $\mathrm{op}_{n,k}(p)$ denote the number of $p$-avoiding ordered set partitions of an $n$-element set into $k$ blocks. For a pattern $p$, write $|p|$ for its length.

Godbole et al.'s conjecture. For each pattern $p$ and each integer $k$, there exists a positive integer $n_0(k,p)$ such that for every $n\geq n_0(k,p)$,

$$\mathrm{op}_{n,k+1}(p)>\mathrm{op}_{n,k}(p)>\cdots>\mathrm{op}_{n,|p|}(p).$$

The conjecture concerns the eventual behavior, as the size of the underlying set grows, of the numbers of pattern-avoiding ordered set partitions. The surrounding results establish asymptotic growth rates for fixed $k$ relative to the pattern length, but do not resolve this eventual monotonicity assertion in general.