Asymptotic monotonicity conjecture for pattern-avoiding ordered set partitions

Let $S_m$ be the symmetric group on $m$ letters, let $\rho\in S_m$ be a permutation, and let $\operatorname{op}_{n,k}(\rho)$ denote the number of ordered set partitions of $[n]$ into $k$ blocks that avoid $\rho$. For each fixed $k$, let $n_0(k)$ be a threshold depending only on $k$. Asymptotic monotonicity conjecture. For each fixed $k$, there exists $n_0(k)$ such that for each $\rho\in S_m$ and $n\ge n_0(k)$,

$$\operatorname{op}_{n,k+1}(\rho)>\operatorname{op}_{n,k}(\rho)>\cdots>\operatorname{op}_{n,\lvert\rho\rvert}(\rho).$$

The conjecture proposes eventual monotonicity in the number of blocks, beginning at $k+1$ and continuing down to the pattern length. The source states that this conjecture was first proved by Marcus and Tardos, so it is recorded as solved.