  We prove that all homology 3-spheres are $J_4$-equivalent, i.e. that any
homology 3-sphere can be obtained from one another by twisting one of its
Heegaard splittings by an element of the mapping class group acting trivially
on the fourth nilpotent quotient of the fundamental group of the gluing
surface. We do so by exhibiting an element of $J_4$, the fourth term of the
Johnson filtration of the mapping class group, on which (the core of) the
Casson invariant takes the value $1$. In particular, this provides an explicit
example of an element of $J_4$ that is not a commutator of length $2$ in the
Torelli group.
