Let $M$ be an oriented smooth compact 3-manifold with non-empty boundary and hyperbolizable interior such that all boundary components have genus greater than $1$. Denote $N:={\rm int}(M)$ and $$HM_{hom}(N):=\{{\rm cocompact~hyperbolic~metrics~on~}N\}/\sim_{hom},$$ where $g_1 \sim_{hom} g_2$ if there exists an orientation preserving diffeomorphism $\phi: N \rightarrow N$ such that $g_1=\phi^*g_2$ and $\phi$ is homotopic to the identity. Cocompact means that the convex core is compact. Also denote $$HM_{iso}(N):=\{{\rm cocompact~hyperbolic~metrics~on~}N\}/\sim_{iso},$$ where the equivalence $\sim_{iso}$ is defined similarly, but $\phi$ is required to be isotopic to the identity.
Let $g \in HM_{hom}(N)$ and $G$ be the image of the corresponding representation of $\pi_1(M)$ in $PSL_2(\mathbb C)$. By $\mathfrak T(G)$ denote the space of quasi-conformal deformations of $G$ modulo conjugacy.
As I understand the discussion in Chapters 5.1-5.2 of Marden's "Hyperbolic manifolds", $HM_{hom}(N)$ can be identified with $\mathfrak T(G)$. Next, Marden states the general version of the Ahlfors-Bers theorem as Theorem 5.1.3: $$\mathfrak T(G)=\prod \mathfrak{Teich}(S_i)/Mod_0(S_i),$$ where $S_i$ are the connected components of $\partial M$ and $Mod_0(S_i)$ is the subgroup of automorphisms of $S_i$ generated by those that have an extension homotopic to the identity in $M$. ($Mod_0(S_i)$ is non-trivial only if $S_i$ is compressible.)
Is it true that $HM_{iso}(N)$ can be identified with just $\prod \mathfrak{Teich}(S_i)$ ? I think that it follows rather directly, but I want to be sure that I am not missing a subtle point.
