The Mamma conjecture for K-theory with dg-category coefficients

Let $G$ be a group, let $R$ be a commutative ring, and let ${\mathcal B}$ be a strictly finite dg cell, meaning a dg category of the finite-cell type specified in the source. For a group $G$, the functor $K(-;{\mathcal B})$ is connective algebraic K-theory with coefficients in ${\mathcal B}$. Let ${\mathcal V}C$ be the family of virtually cyclic subgroups of $G$, and let the ${\mathcal V}C$-assembly property refer to assembly for this family. A ring is regular when it is noetherian and has finite projective dimension.

Mamma conjecture. Given a group $G$, the functors $K(-;{\mathcal B})$ have the ${\mathcal V}C$-assembly property for all strictly finite dg cells ${\mathcal B}$, when the base ring $R$ is regular and the orders of all finite subgroups of $G$ are invertible in $R$.

By the theorem immediately preceding the conjecture, this formulation is equivalent to the corresponding assertion for the fundamental additive invariant and also to the formulation using all homotopically finitely presented dg categories. It reduces the proposed isomorphism statement to K-theory with finite dg-cell coefficients, but the conjecture is not resolved in the supplied source.