The classification conjecture for surjective rank-k projector maps when n=2k

Let $\mathcal{H}$ be an $n$-dimensional Hilbert space, and let $\mathcal{P}_k(\mathcal{H})$ denote the set of rank-$k$ orthogonal projectors on $\mathcal{H}$. Assume that $n=2k$ and that $\Phi:\mathcal{P}_k(\mathcal{H})\to\mathcal{P}_k(\mathcal{H})$ is surjective. Let $R_k$ be the previously defined complementary-projector map, and let the maps of the form in equation be the standard maps under consideration. Classification conjecture. Every such surjective map is either a map of the form in equation or the composition of one of those maps with $R_k$, namely $\Phi\circ R_k$. This conjecture concerns the remaining classification problem for surjective endomorphisms of the rank-$k$ projector space in the case $n=2k$, where the complementary-projector construction gives additional maps beyond the standard form. The supplied text does not state a resolution, so the conjecture is recorded as open.