Complementation conjecture for Wilf equivalence of pattern pairs

Let $\pi_1 otin\Pi_k$ and $\pi_2 otin\Pi_l$ with $k otin l$. For a pattern $\pi$, write $\pi^c$ for its complement, and let $\Pi_l(\pi_1)$ denote the set of length-$l$ patterns avoided jointly with $\pi_1$. Complementation conjecture.

1. If $\pi_2,\pi_2^c otin\Pi_l(\pi_1)$, then $(\pi_1,\pi_2)$ is Wilf equivalent to $(\pi_1,\pi_2^c)$.
2. If $\pi_2 otin\Pi_l(\pi_1)$ and $\pi_2^c otin\Pi_l(\pi_1)$, then $(\pi_1,\pi_2)$ is Wilf equivalent to $(\pi_1^c,\pi_2^c)$.

The conjecture proposes symmetries between avoidance classes of pairs of set-partition patterns, motivated by cardinality computations for several $(3,4)$- and $(4,4)$-pattern pairs. Its general validity is not established in the supplied text.