Drużkowski's invertibility conjecture for cubic maps

Let $n\geq 1$ and let $A=\{a_{ij}\}_{1\leq i,j\leq n}$ be an $n\times n$ complex matrix. Define the map $T:\mathbb{C}^n\to\mathbb{C}^n$ by

$$w_i=z_i+g_i^3,\qquad g_i=a_{i1}z_1+a_{i2}z_2+\cdots+a_{in}z_n,\quad 1\leq i\leq n.$$

Such a map is called a Drużkowski map. Drużkowski's conjecture. If the Jacobian determinant $J(T)$ is a nonzero constant, then $T$ is invertible. This is a reduction of the Jacobian Conjecture: Drużkowski showed that proving this assertion for all $n\geq 1$ would imply the Jacobian Conjecture.