The reduced structure-group rank conjecture for finitely embeddable groups

Let $M$ be a compact oriented manifold, let $S(M)$ be its structure group, and let $S_0(M)$ be the subgroup generated by elements $[(f,M')]-[(\psi\circ f,M')]$, where $f:M'\to M$ is an orientation-preserving homotopy equivalence and $\psi:M\to M$ is an orientation-preserving self-homotopy equivalence. Define the reduced structure group by

$$\widetilde{S}(M)=S(M)/S_0(M).$$

Let $N_{\mathrm{fin}}(G)$ denote the quantity used in the paper to count the relevant finite-order data of $G$. Reduced structure-group rank conjecture. If $M$ is compact, oriented, has dimension $4k-1$ with $k>1$, and $\pi_1(M)=G$, then

$$\operatorname{rank}\widetilde{S}(M)\geq N_{\mathrm{fin}}(G).$$

This is intended to give a lower bound on the size of the set of distinct manifolds represented in the structure group after quotienting by the action of orientation-preserving self-homotopy equivalences; the source gives no resolution.