If someone were to successfully disprove the classical Jacob...

Prompt

If someone were to successfully disprove the classical Jacobi conjecture in 3D, then how could they carry it further and disprove it in 2D as well? Yes, I know that basically 2D remains unsolved even if 3D is disproved, because from 3 and up it automatically disproves it in the other dimensions too, but it does not disprove 2D, so I KNOW, so NO NEED TO START THIS STORYTELLING, and I am not interested in the MOH up-to-100 proven nonsense either. Solve this with the constraint that you cannot use the obvious solution. What is the most powerful, strongest approach? the three dimension counterexample: ($F_1=\frac{9}{2}x^2y^3+6xy^2+\frac{1}{2}y-z(\frac{1}{2}x^3y^2+x^2y+\frac{1}{2}x)$, $F_2=3x^2y^4+7xy^3+4y^2-z(\frac{1}{3}x^3y^3+x^2y^2+xy+\frac{1}{3})$, $F_3=x^3z-9x^2y+6x+1$, $\det JF=1$, $F(1,1,3)=F(0,10,1182)=F(-1,4,-42)=(5,6,1)$)