MMS-type problems for Johnson scheme

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.