The strict ordering of non-one-block patterns

Let $\Pi_k$ be the set of partitions of $[k]$, let $\beta_k\in\Pi_k$ be the partition with one block, and let $\Pi_n(\tau)$ denote the set of partitions of $[n]$ avoiding $\tau$. Define $\tau\prec\pi$ when $|\Pi_n(\tau)|\leq |\Pi_n(\pi)|$ for all $n>k$, with strict inequality for all sufficiently large $n$.

Strict ordering conjecture. If $k\geq 4$, $\tau\in\Pi_k$, and $\tau\neq\beta_k$, then

$$\tau\prec\beta_k$$

and

$$|\Pi_n(\tau)|<|\Pi_n(\beta_k)|$$

for all $n>k$.

Computer evidence suggests this ordering, while the paper proves it for patterns with exactly two blocks; the assertion for all other patterns remains conjectural in the supplied text.