The quotient conjecture for equivariant K-theory

Let $X$ be a noetherian regular separated algebraic space over a field $k$, and let $G$ be a linear algebraic $k$-group acting on $X$ with finite stabilizers such that the quotient $X/G$ exists as a regular algebraic space. Let $N$ be the least common multiple of the orders of all the essential dual cyclic subgroups of $G$, and set

$$\Lambda={\mathbb Z}\left[1/N\right].$$

If $p:X\@ifnextchar^ {\t@@}{\t@@^{}} X/G$ is the quotient map, consider the composition

$$K_{*}(X/G)_\Lambda\stackrel{p^{*}}{\longrightarrow}K_{*}(X,G)_\Lambda\longrightarrow K_{*}(X,G)_{{\rm geom}}.$$

Quotient conjecture. This composition is an isomorphism.

The conjecture expresses that geometric equivariant K-theory should agree with the K-theory of the quotient when the quotient is regular, after inverting the orders of all essential dual cyclic subgroups of $G$.