Sorry, that's incorrect.

The net torque on the top about the point where the top contacts the pedestal causes the top's precession. The precessional angular speed ${}_p$ is a function of the top's moment of inertia I, the top's mass $m_{\text{t}\text{o}\text{p}},$ the top's initial angular speed ${}_0,$ and the distance $r_{cm}$ from the top's center of mass to the point where the top contacts the pedestal.

${}_p=\frac{m_{\text{t}\text{o}\text{p}}g{r}_{cm}}{I{}_0}$

The location of the center of mass of a composite object depends on the centers of mass of the object's constituent pieces.

Similarly, the moment of inertia of a composite object depends on the moments of inertia of the object's constituent pieces. Assume the thin cylinder has a negligible radius, and consider how this affects the total moment of inertia of the top.