Deflatability conjecture for parallel alternations

Let $pi$ be a parallel alternation, meaning a permutation whose entries alternate between two monotone subsequences, with $|pi|\geq 6$. For a permutation $pi$, let $\operatorname{Av}(pi)$ denote the class of permutations avoiding $pi$.

Deflatability conjecture. The class $\operatorname{Av}(pi)$ is deflatable.

This conjecture is motivated by explicit deflatability witnesses for several parallel alternations, including $246135$ and longer examples. The general assertion for all parallel alternations of length at least six remains open in the supplied source.