Determine the source of stretched-exponential asymptotics

Determine what structural features determine whether a class of pattern-avoiding inversion sequences or permutations exhibits stretched-exponential asymptotic behavior, including the conjectured $\mu_1^{n^{3/8}}$ factor observed for Classes 247 and 759.

Background

The paper reports unusually complex asymptotic estimates for Classes 247 and 759, whose counting sequences are conjectured to contain stretched-exponential factors of the form $\mu_1^{n^{3/8}}$. This behavior is compared with conjectured asymptotics for 1324-avoiding permutations.

The authors explicitly identify as unresolved the general question of which pattern-avoidance classes exhibit such behavior. Resolving it would provide a structural criterion distinguishing ordinary algebraic-type asymptotics from stretched-exponential growth.