Extension of orbitmesy classification beyond Z_4

Establish whether an orbitmesy characterization analogous to the Z_4 result exists for promotion orbits of increasing labelings of the zig-zag poset Z_n with n even and n different from 4, beginning with Z_6, with respect to the antipodal sum statistic.

Background

The paper completely characterizes the promotion orbits of increasing labelings of Z_4 that are orbitmesic with respect to antipodal sum, using pattern avoidance and a balancedness condition. The authors note that antipodal sum is naturally restricted to even-sized zig-zag posets.

For Z_6, the paper identifies some orbitmesic families through the linear-extension and stability criteria, but it does not provide a complete classification. The open question asks whether the structural characterization proved for Z_4 has an analogue for larger even n.