Fausk–Lewis–May conjecture on homotopy representations

Let $G$ be a compact Lie group. A homotopy representation of $G$ is a finite-dimensional $G$-representation sphere-type object, and let $V(G)$ denote its Grothendieck group. Let $V'(G)$ denote the Grothendieck group of generalized homotopy representations. There is a canonical map

$$V(G)\xrightarrow{}V'(G).$$

Fausk–Lewis–May conjecture. The canonical map

$$V(G)\xrightarrow{}V'(G)$$

is an isomorphism for any compact Lie group $G$. This conjecture concerns the comparison between homotopy and generalized homotopy representations and, via the identification of $V'(G)$ with the Picard group of the equivariant stable homotopy category, gives structural information about invertible $G$-spectra. The paper's abstract states that the conjecture is resolved for compact Lie groups whose component group has all projective integral group-ring modules stably free; it does not establish the assertion for every compact Lie group.