The Mamma conjecture for the fundamental additive invariant

Let $G$ be a group and let $R$ be a commutative ring. A ring is regular when it is noetherian and has finite projective dimension. Let ${ obreak\boldsymbol{\mathbf{E}}}^{\mathsf{fund}}$ denote the fundamental additive invariant on the orbit category of $G$, and let ${\mathcal V}C$ be the family of virtually cyclic subgroups of $G$. The ${\mathcal V}C$-assembly property is the assembly property for this family.

Mamma conjecture. Given a group $G$, the fundamental additive invariant ${\nobreak\boldsymbol{\mathbf{E}}}^{\mathsf{fund}}$ has the ${\mathcal V}C$-assembly property when the base ring $R$ is regular and the orders of all finite subgroups of $G$ are invertible in $R$ (for example, when $R$ is a regular $\mathbb{Q}$-algebra).

This is proposed as a common source of many isomorphism conjectures. The stated conditions remove the Bass–Heller–Swan obstruction that occurs for arbitrary rings, but the conjecture remains unproved in the supplied source.