Let $M$ be a closed hyperbolic 3-manifold and $H_{g}$ a genus g handlebody. Assume that $\pi: int(H_{g})\rightarrow M$ is a regular cover. Denote $N\subset H_{g}$ the convex core.

My question is: If the diameter of $\pi(\partial N)$ is finite in $M$, is it true for $\pi(N)$ in $M$?

Thank you!