CR refers to *Methods of Representation Theory* by Charles Curtis and Irving Reiner.

Let $F$ be a finite extension of $\mathbb{Q}_p$ with valuation ring $\mathcal{O}_F$. Let $G$ be a finite group and let $A$ be the group algebra $F[G]$. Let $\Lambda$ be an $\mathcal{O}_F$-order in $A$. Let $K_0(\Lambda)$ (resp. $K_0(A)$) denote the Grothendieck group of the category of finitely generated projective left $\Lambda$-modules (resp. $A$-modules).

Let $SK_0(\Lambda)=\ker \varphi$ where $\varphi: K_0(\Lambda) \longrightarrow K_0(A)$ is the map given by $\varphi([M])=[F \otimes_{\mathcal{O}_F} M]$ (this definition is taken from CR vol 2, top of page 222.)

Is $SK_0(\Lambda)$ always trivial? I know that this is true in the following cases:

- $\Lambda=\mathcal{O}_{F}[G]$ - see CR vol 1, Theorem 32.1
- more generally, when $\Lambda$ satisfies the conditions of CR vol 1, Theorem 32.5
- $\Lambda$ is a maximal order - see CR vol 1, Theorem 26.24iii
- $\Lambda$ is commutative - see CR vol 1, Proposition 35.7

Can anyone provide (a reference to) a proof of the general case or counterexample?