Indicator-dimension conjecture for polyhedral groups

For a module $M$ over a poset $Q$, define its downset-dimension and upset-dimension as the smallest lengths of downset and upset resolutions, respectively. Define the indicator-dimension of $M$ as the maximum of these two dimensions, and define the indicator-dimension of $Q$ as the maximum indicator-dimension of its tame modules.

Indicator-dimension conjecture. The indicator-dimension of any real or discrete polyhedral group $Q$ equals the rank of $Q$ as a free module, over the field $\mathbb{R}$ or the group $\mathbb{Z}$, respectively.

The conjecture proposes that the geometry and rank of a polyhedral group control the lengths of indicator resolutions, analogously to the Hilbert Syzygy Theorem. The source presents this as an open direction and notes that proving it would require developing a derived-functor theory for indicator resolutions.