  In the current work we consider the minimization problems for the number of
nonzero or negative values of vectors from the first and second eigenspaces of
the Johnson scheme respectively. The topic is a meeting point for
generalizations of the Manikam-Mikl\'{o}s-Singhi conjecture proven by Blinovski
and the minimum support problem for the eigenspaces of the Johnson graph,
asymptotically solved by authors in a recent paper.
