  We consider a higher order in (time) semilinear evolution inequality posed on
the Kor\'{a}nyi ball under an inhomogeneous Dirichlet-type boundary condition.
The problem involves an inverse-square potential $\lambda/|\xi|_\mathbb{H}^2$,
where $\lambda \geq -(Q-2)^2/4$ and a general weight function $V$ depending on
the space variable in front of the power nonlinearity. We first establish a
general nonexistence result for the considered problem. Next, in the special
case $V(\xi):=|\xi|_\mathbb{H}^a$, $a\in \mathbb{R}$, we prove the sharpness of
our nonexistence result and show that the problem admits three different
critical behaviors according to the value of the parameter $\lambda$.
