Answer
NOT conservative
Work Step by Step
The vector field $F$ is conservative when $curl F=0$
When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$
$curl F=(6x^2yz^2-6x^2yz^2)i+(6xy^2z-6xy^2z^2)j+(4xyz^3-6xyz^2)k \ne 0$
Thus, the vector field $F$ is NOT conservative because $curl F \ne 0$.
