The strongly-algebraic characterization conjecture for permutation classes

Let $\mathcal{C}$ be a permutation class. It is strongly algebraic if $\mathcal{C}$ and every subclass of $\mathcal{C}$ have algebraic generating functions, and it is well-quasi-ordered if it contains no infinite antichain. The strongly-algebraic characterization conjecture. A permutation class is strongly algebraic if and only if it is well-quasi-ordered. Strongly algebraic classes are necessarily well-quasi-ordered, while the paper's main result gives well-quasi-ordered classes that are not algebraic, refuting the converse and hence the biconditional.