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 ≥ 1 obtained in [ES2] and a hook-type component [3, 1 5 ] detected in [EE] appear in their gap.