Existence of 3-inflatable permutations in every admissible residue class modulo 144

A permutation is 3-inflatable if its pattern densities of length $3$ satisfy the prescribed inflation criterion. The admissible residue classes modulo $144$ are the residue classes identified by the necessary conditions for the existence of such permutations.

Existence conjecture. For every positive integer $x$ whose length belongs to an admissible residue class modulo $144$, there exists a 3-inflatable permutation of length $x$.

The conjecture is motivated by computer searches that found centrally symmetric 3-inflatable permutations of lengths $17$, $64$, $80$, $81$, $144$, and $145$, as well as examples in every admissible residue class modulo $144$.