Relative Farrell–Jones conjecture for principal series Bernstein blocks

Establish that, for a connected split reductive p-adic group G of rank one and any principal series type ρχ constructed via Roche’s method, the relative assembly map Colim_{H ∈ Orb^∞_Cop(G)} K(H, ρχ) → K(G, ρχ), formed by restricting the Farrell–Jones assembly to the subcategories D(H, ρχ) closed under compact induction and defining K(H, ρχ) = K(D(H, ρχ)^ω), is an equivalence of spectra.

Background

To connect K-theory of Bernstein blocks with compact open subgroups, the authors associate to each principal series type (J, ρχ) a compatible system of subcategories D(K, ρχ) ⊂ D(K) that is closed under compact induction. This enables a "relative" assembly map analogous to the global Farrell–Jones map but targeted at the K-theory of a single Bernstein block.

They prove rank-one split cases (Theorem B) yield a pushout/cofiber description, but explicitly conjecture that the entire relative assembly map is an equivalence. This is a natural strengthening of the global assembly paradigm at the level of Bernstein components.