The cubic homogeneous nilpotent-Jacobian formulation of the Jacobian conjecture

Let $H(z)$ be a homogeneous polynomial map of degree $d=3$, let $JH(z)$ denote its Jacobian matrix, and let $N_t(z)$ be the unique solution of the Cauchy problem

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Cubic nilpotent-Jacobian conjecture. If $JH(z)$ is nilpotent, then $N_t(z)$ is a polynomial solution in both $t$ and $z$. This is presented as an equivalent formulation of the Jacobian conjecture after the reduction theorem cited in the source. The source notes that the cases with nilpotency index at least three are substantially more difficult than the square-zero case, while the cubic cases in dimensions three and four were known in the cited special results. The status of the general formulation remains open.