Klein–Williams conjecture on equivariant Nielsen numbers

Let $G$ act on a closed smooth manifold $M$, let $f:M\to M$ be a $G$-equivariant map, and let $\ell_G(f)$ be the Klein–Williams invariant. For each conjugacy class of subgroups $H$ of $G$, the tom Dieck splitting decomposes the relevant equivariant framed bordism group into a corresponding summand, and $f^H$ denotes the induced map on the fixed set $M^H$. Klein–Williams conjecture. The Nielsen numbers $N(f^H)$ can be computed from the projection of $\ell_G(f)$ onto the corresponding summand in the tom Dieck splitting. This conjecture is presented as the main motivation of the paper; the supplied text does not establish whether it has been resolved.