The conjecture that d-complete posets are LE-cactus

A d-complete poset is a poset in the sense of the standard definition of d-completeness. A poset is LE-cactus when the Bender–Knuth involutions acting on its linear extensions satisfy the cactus group relations, including $(t_iq_{jk})^2=1$ for $i+1<j<k$, where $q_{jk}=q_{k-1}q_{k-j}q_{k-1}$ and $q_i=t_1(t_2t_1)\cdots(t_it_{i-1}\cdots t_1)$. The d-complete poset conjecture. Every d-complete poset is LE-cactus. The conjecture extends the known LE-cactus families, which include Ferrers posets, shifted Ferrers posets, rooted trees, and other minuscule posets; its resolution is not specified in the supplied source.