The even zigzag-poset LE-symmetry conjecture

For even $n$, let $Z_n$ be the poset on $n$ elements with relations

$$v_1 < v_2 > v_3 < v_4 > \ldots > v_{n-3} < v_{n-2} > v_{n-1} < v_n.$$

A poset is LE-symmetric when its Bender–Knuth group acts as the full symmetric group on its set of linear extensions. The even zigzag-poset LE-symmetry conjecture. The zigzag-poset $Z_n$ is LE-symmetric when $n$ is even. This is computationally verified for all even $n\leq 10$, while odd cases such as $Z_5$, $Z_7$, and $Z_9$ provide counterexamples; the even case remains open.