Nilpotency and inversion-step conjecture for cubic homogeneous perturbations

Let $F:K^d\rightarrow K^d$ be a polynomial map with $F_i=X_i+H_i$, where

$$H_i=L_i^3,$$

and

$$L_i=a_{i1}X_1+\dots+a_{id}X_d,$$

for $1\leq i\leq d$. Write $H=(H_1,\dots,H_d)$ and let $JH$ be its Jacobian matrix. Define polynomial sequences $(P_j^i)$ by

$$P_0^i=X_i,\qquad P_1^i=H_i,$$

and, whenever $P_{j-1}^i$ is defined,

$$P_j^i(X_1,\dots,X_d)=P_{j-1}^i(F_1,\dots,F_d)-P_{j-1}^i(X_1,\dots,X_d).$$

Let $g$ be an integer with $1\leq g\leq d$. Nilpotency and inversion-step conjecture. If $(JH)^g=0$, then

$$P_{(3^{g-1}+1)/2}^i=0$$

for every $i=1,\dots,d$; moreover, $F$ is invertible and its inverse has maximal degree at most $3^{g-1}$.

This conjecture predicts that nilpotency of the Jacobian controls both the termination of the inversion algorithm and the degree of the inverse. The supplied text formulates it after proving a special case, but gives no resolution evidence for the general statement.