Analytical Whitehead conjecture

Prove that if a discrete group G admits a 2-dimensional model for BG, then every invertible matrix over C_r^*(G) is stably homotopic to a diagonal matrix whose first diagonal entry is an element of G and whose remaining diagonal entries are 1.

Background

This conjecture is an analytic analogue of a Whitehead-group vanishing statement. It concerns the stable homotopy classes of invertible matrices over the reduced group C*-algebra. The paper states that it follows from surjectivity of the Baum–Connes assembly map when G has a 2-dimensional classifying space, and identifies it with the vanishing of an analytical Whitehead group.

References

Assume that $G$ admits a 2-dimensional model for $BG$. Then any invertible $T\in GL_n(A)$ is stably homotopic to a $N$-by-$N$ diagonal matrix $$\left(\begin{array}{ccccc}g & 0 & 0 & \cdots & 0 \0 & 1 & 0 & \cdots & 0 \0 & 0 & 1 & \cdots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \0 & 0 & 0 & \cdots & 1\end{array}\right)$$ for some $N\geq n$ and some $g\in G$.

— The Baum-Connes conjecture: a concise course  (2610.01802 - Valette, 1 Oct 2026) in Conjecture 4, Section 3.2, “A Whitehead group-type conjecture”