Klein–Williams conjecture on the terms of the equivariant invariant

Let $M$ be a closed smooth manifold, let $f:M\to M$ be a self-map, and let $\ell_n(f)$ lie in the direct sum $\bigoplus_{k\mid n}\mathbb{Z}[\pi_1(M)_{\rho,k}]$, where $\pi_1(M)_{\rho,k}$ is the basis set described by the equivalence relations in the source. Let $\mathcal{N}_k(f)$ be the number of non-zero terms in the component of $\ell_n(f)$ expressed as a linear combination of the basis elements of $\mathbb{Z}[\pi_1(M)_{\rho,k}]$. Klein–Williams conjecture. For every divisor $k$ of $n$,

$$\mathcal{N}_k(f)=N(f^k).$$

This is the proposed assertion that the Klein–Williams invariant contains at least as much information as the Nielsen numbers of the iterates. The supplied text does not state whether this conjecture has been proved or disproved.