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 for $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see [this paper][1] Corollary 4.33). It's also true (using the function [is_cohen_macaulay][2] on SAGE) for the three first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed [here][3] (my desktop isn't enough powerful   for checking the fourth one).


  [1]: http://arxiv.org/pdf/1604.06765v4.pdf
  [2]: http://combinat.sagemath.org/doc/reference/homology/sage/homology/simplicial_complex.html
  [3]: https://mathoverflow.net/q/245090/34538