Rousseeuw–Hubert partition conjecture for regression depth

Let $P$ be a set of $n$ points, and let regression depth be the minimum number of points intersected by a hyperplane during any continuous motion taking it to a vertical hyperplane. A hyperplane has nonzero regression depth in a subset when it has positive regression depth with respect to that subset. Rousseeuw–Hubert partition conjecture. There exists a partition of $P$ into $\lceil n/(d+1)\rceil$ subsets and a hyperplane that has nonzero regression depth in each subset. This is a strengthening in partition form of the regression-depth questions motivated by center-point analogies; the supplied source does not state whether it has been resolved.