The genuine equivariant K-theory block decomposition conjecture

Let $G$ be a finite group, let $p$ be a prime, and let $q$ denote an unramified extension parameter. Write $\mathbf{K}_{G,q}$ for the corresponding unramified genuine equivariant K-theory spectrum, and $\mathbf{K}_q[\mathit{Z}_G(c)]$ for the K-theory group-ring spectrum of the centralizer $\mathit{Z}_G(c)$ of an element $c\in G$. The notation $(-)^\omega$ denotes compact objects and $(-)^{\mathit{ft}}$ denotes the finite-type subcategory.

Genuine equivariant K-theory block decomposition conjecture. There is an equivalence of stable $\infty$-categories

$$\mathrm{LMod}(\mathbf{K}_{G,q})^\omega \cong \bigoplus_c \mathrm{LMod}(\mathbf{K}_q[\mathit{Z}_G(c)])^{\mathit{ft}},$$

where the sum is over representatives for conjugacy classes of elements $c\in G$ of order prime to $p$, and $\mathit{Z}_G(c)$ denotes the centralizer of $c$.

The conjecture proposes that the additional blocks appearing in genuine equivariant K-theory are not essentially new, and is closely related to a question raised by Akhil Mathew. It concerns the Bredon, rather than Borel, equivariant theory; the supplied source does not state whether it has been resolved.