The n-GCD conjecture for polynomial automorphisms

Let $F:\mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$ be a $\mathbb{C}$-algebra endomorphism with invertible Jacobian. Write $F_i=F(x_i)$, and let $l_i$ be the degree of $F_i$ for $1\leq i\leq n$. The $n$-GCD conjecture. If

$$\gcd(l_u,l_v)=1$$

for every pair $1\leq u\ne v\leq n$, then $F$ is an automorphism. This extends the two-variable Magnus-type results to higher dimensions, but the supplied text gives no resolution of the conjecture.