  In [ES2], the first and the third authors introduced new classes in the
Johnson cokernels of the mapping class groups of surfaces by a representation
theoretic approach based on some previous results for the Johnson cokernels of
the automorphism groups of free groups. On the other hand, in [KK1], Kawazumi
and the second author introduced another type of classes by a topological
consideration of self-intersections of curves on a surface.
  In this paper, we show that the classes found in [KK1] are contained in the
classes found in [ES2] in a stable range. Furthermore, we prove that the
anti-Morita obstructions $[1^{4m+1}]$ for $m \ge 1$ obtained in [ES2] and a
hook-type component $[3,1^5]$ detected in [EE] appear in their gap.
