Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

*Question*: Is $\hat{C}(H,G)$ Cohen-Macaulay?

*Remark*: It is true if $|G:H|<32$ (see Corollary 4.33 of [this paper][1]). It's also true for the four first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed [here][2].


  [1]: http://arxiv.org/pdf/1604.06765v4.pdf
  [2]: http://mathoverflow.net/q/245090/34538