Finite dense resolutions for tame modules over real polyhedral groups

Let $Q$ be a real polyhedral group, and let $M$ be a tame, semialgebraic, or PL module over $Q$. A dense downset or upset resolution is a resolution of the corresponding type whose terms satisfy the density condition for the real polyhedral setting.

Finite dense resolution conjecture. Every tame, semialgebraic, or PL module $M$ over a real polyhedral group $Q$ has finite dense downset and upset resolutions of the corresponding type.

Dense resolutions can be constructed from scratch, but the source notes that there is no a priori guarantee that these constructions terminate after finitely many steps. Thus the finiteness assertion remains open.