The indicator-dimension conjecture for real and discrete polyhedral groups

Let $Q$ be a real or discrete polyhedral group, and define the indicator-dimension of $Q$ as the maximum of the indicator-dimensions of its finitely encoded modules, where the indicator-dimension of a module is the maximum of its downset- and upset-dimensions.

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

This conjecture proposes a Hilbert-syzygy-type bound for indicator resolutions, relating their lengths to the rank of the underlying polyhedral group rather than to the order dimension of an encoding poset. The surrounding discussion presents it as an open direction motivated by the absence of a general finite-termination guarantee for minimal indicator resolutions.