Finiteness and unboundedness for Turaev–Viro classes of hyperbolic fibered 3-manifolds

For a closed 3-manifold $M$, let ${\mathcal X}^{TV}(M)$ denote the set of homeomorphism classes of closed orientable 3-manifolds having the same Turaev–Viro invariants as $M$. Let $g\geq 1$ be the genus of a fiber. Turaev–Viro fibered-manifold question. The number of homeomorphism classes in ${\mathcal X}^{TV}(M)$ represented by hyperbolic fibered 3-manifolds $N$ with fiber of genus $g$ should be finite for every $M$; is this number unbounded as $M$ ranges over hyperbolic fibered 3-manifolds with fiber of a given genus?

This asks whether the unboundedness established for torus bundles extends to higher-genus hyperbolic fibered 3-manifolds. The source presents the finiteness assertion together with the unboundedness question, without resolving either in general.