Weinberger's homotopy invariance conjecture for the rho-invariant

Let $M$ be a closed $(2k-1)$-manifold with torsion-free fundamental group, and let $\alpha$ be a unitary representation used to define the rho-invariant $\rho(M,\alpha)$. Weinberger's conjecture. The invariant $\rho(M,\alpha)$ depends only on the homotopy type of $M$. Neumann proved this for free abelian fundamental groups, and the Farber–Levine–Weinberger theorem covers groups whose $U(n)$ character varieties are connected, such as free groups. Wall's calculations for lens spaces disprove the extension to all groups, so the conjecture as stated is refuted.