Answer
NOT conservative.
Work Step by Step
The vector field $F$ will be conservative if and only if $curl F=0$
Let us consider $F=P i+Q j+R k$
$curl F=[R_y-Q_z]i+[P_z-R_z]j+[Q_x-P_y]k$
Now, $curl F=(0-12x^2yz^2)i+(4xyz^3-2xy z^4)j+(8xyz^3-xz^4) \ne 0$
Thus, the vector field $F$ is NOT conservative.