The irreducibility of symmetric Yagzhev maps

Let $F:\mathbb{C}^n \rightarrow \mathbb{C}^n$ be a polynomial mapping in Yagzhev form, i.e.

$$F(x_1,\ldots,x_n)=(x_1+H_1(x_1,\ldots,x_n),\ldots,x_n+H_n(x_1,\ldots,x_n)),$$

where $H_i$ are homogeneous polynomials of degree 3. We show that if $\mathrm{Jac}(F) \in \mathbb{C}^*$ and the Jacobian matrix of $F$ is symmetric, then the polynomials $x_i+H_i(x_1,\ldots,x_n)$ are irreducible as elements of the ring $\mathbb{C}[x_1,\ldots,x_n]$.