Answer
NOT conservative.
Work Step by Step
The vector field $F$ will be conservative if and only if $curl F=0$
consider $F=A i+B j+C k$
$curl F=[C_y-B_z]i+[A_z-C_z]j+[B_x-A_y]k$
Here, we have $curl F=(2x^2yz -2x^2yz)+(2xyz-2xy^2 z)+(2xyz^2-xz^3) \ne 0$
Thus, the vector field $F$ is NOT conservative.
