The Ferrers-poset LE-symmetry conjecture for specified partitions

Let $F_\lambda$ be the Ferrers poset associated with a partition $\lambda$, and let $\lambda^t$ be the conjugate partition; the posets $F_\lambda$ and $F_{\lambda^t}$ are isomorphic. A poset is LE-symmetric when its Bender–Knuth group acts as the full symmetric group on its set of linear extensions. The Ferrers-poset LE-symmetry conjecture. The Ferrers posets $F_\lambda$, and therefore $F_{\lambda^t}$, are LE-symmetric in the following cases: $\lambda=(n,n-2)$ for all $n$; $\lambda=(n,3)$ for $n\not\equiv 2\pmod 4$; and $\lambda=(n,2,2)$ for $n\not\equiv 0\pmod 4$. These are conjectural families motivated by the contrast between LE-cactus and LE-symmetric behavior; the stated cases remain open.