About the Jacobian conjecture counterexample and the determinant

So recently the Jacobian conjecture has been disproven.

The counterexample had a determinant of $-2$.

See for instance :

https://www.newscientist.com/article/2580374-ais-solution-to-87-year-old-riddle-takes-mathematicians-by-surprise/

or Wikipedia.

Now I wonder if this polynomial can lead to an example where the determinant is $2$. Can we flip the sign ?

In fact I read on the wiki page that :

(quote) Edwin Connell and Lou van den Dries proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant $1$. (unquote)

So I was surprised to read the example had a negative determinant.

Can we construct an example with determinant $1$ based on the known counterexample and staying in dimension $3$ ?

Does a "smart" substitution of variables do it ?

And if so, why the "fuss" about determinant $1$ ?

I am confused.

I know that a positive determinant implies locally orientation-preserving and a negative one implies locally orientation-reversing. And since in the Jacobian conjecture the sign never changes this implies neither does the orientation.

But that does not fully clarify it for me.

Maybe it relates to the fact that we have $3$ points mapping to $1$ for the example given by the AI. Maybe if we get a $2$ points to $1$ mapping ???

I was thinking about Resultants and Grobner basis and such but that is probably going in the wrong direction and way to complicated.