Finite minimal resolutions for tame modules over discrete polyhedral groups

Let $Q$ be a discrete polyhedral group, and let $M$ be a tame module over $Q$. A minimal downset or upset resolution is a resolution of the corresponding type with no redundant indicator summands.

Finite minimal resolution conjecture. Every tame module $M$ over a discrete polyhedral group $Q$ has finite minimal downset and upset resolutions of the corresponding type.

The source contrasts this with the existence of dense resolutions in the real polyhedral setting and states that finiteness of minimal indicator resolutions is not guaranteed by the available constructions. The assertion therefore remains open.