Answer
The assertion is True.
Work Step by Step
The moment of inertia about the $z$-axis of a solid $E$ with constant density $k$ is
$$
\begin{aligned}
I_{z}&=\iiint_{E}\left(x^{2}+y^{2}\right) \rho(x, y, z) d V \\
&=\iiint_{E}\left(k r^{2}\right) r d z d r d \theta \\
&=\iiint_{E} k r^{3} d z d r d \theta
\end{aligned}
$$
So the assertion is True.