Palindromicity and gamma-positivity under symmetric avoidance

Characterize the symmetric pattern sets Λ for which the descent polynomial of canon permutations avoiding Λ is palindromic or γ-positive.

Background

For the symmetric avoidance family (1^a21^b), the paper proves that the descent polynomial agrees with the descent polynomial D_n^ℓ(x) for a smaller canon class, and therefore is palindromic and γ-positive.

The paper asks for a general characterization. Through the lattice-word reduction, each symmetric avoidance class can be viewed as a union of linear-extension sets of labeled posets obtained from the rectangle poset [k] × [n] by adjoining relations imposed by avoidance. The existing poset machinery applies when the resulting poset retains a product structure, but the general situation remains unresolved.