Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is the set $\{ \operatorname{Vol}(\rho) : \rho \in \mathfrak{R}_0(M) \}$ discrete? Or finite? Does the answer change with the number of cusps of $M$, or get simpler if $M$ is a link or knot complement? (Clearly the restriction to boundary-parabolic representations is critical here, because otherwise we get volumes of hyperbolic Dehn fillings approaching $\operatorname{Vol}(M)$ from below.)
Proof idea: The volume is the real part of the complex Chern-Simons invariant. This is known to be constant on connected components of $\mathfrak{R}_0(M)$, of which there are finitely many, so in particular the volume can take finitely many values. Does this work? Is there a more direct proof?
