I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem by Sullivan, which states that if a 3-manifold $M$ allows at least one hyperbolic structure, there is a 1-1 correspondence between hyperbolic structures on $M$ and conformal structures on $\partial M$.
I want to apply this theorem to the figure-8 knot complement $M=S^3- K$, where $K$ denotes the figure-8 knot. The boundary of $M$ is $\partial M = T^2$. Hence, its conformal structure can be parametrized by a complex number $\tau$ (modulo $SL(2,Z)$). My question is does Sullivan's theorem mean that for any choice of $\tau$ there is a corresponding hyperbolic structure on $M$ (regardless of whether the volume of the $M$ is finite or not)?
Thurston's note constructs a hyperbolic structure for the figure-8 knot complement $M=S^3- K$ by gluing two tetrahedra together. The $\tau$ parameter for the boundary conformal structure seems to be uniquely fixed (to be some third root of unit?) in this construction. And the hyperbolic volume of $M$ is finite. If the answer to my question is yes, does it mean that, if $\tau$ deviates from the values given by Thurston's construction, $M$ can still have a hyperbolic structure but the volume has to be infinite? If the answer to my question is no, then what exactly is the statement of Sullivan's theorem?