Polyhedral seminorm conjecture for twisted $L^2$-Betti numbers

Let $G$ be a finitely generated group satisfying the Atiyah conjecture, and let $C_*$ be a finitely generated free $L^2$-acyclic $\mathbb{Z}G$-chain complex. For a cohomology parameter represented by the placeholder ${-}$, let $b_n^{(2)}(C_*;{-})$ denote the corresponding twisted $L^2$-Betti number. Polyhedral seminorm conjecture. For every $n\ge 0$, the function

$$b_n^{(2)}(C_*; {-})$$

is a polyhedral seminorm. This generalizes the seminorm assertion in the twisted Singer conjecture and is intended as a step toward proving it; no resolution is supplied in the given text.