Bounded normalizer conjecture for finite-volume hyperbolic 3-manifold groups

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subgroup of $\pi_1(N)$ isomorphic to $\mathbf{Z}\oplus\mathbf{Z}$ is peripheral, and that $N$ is not an $I$-bundle over a torus or Klein bottle. For a subgroup $H\leq\pi_1(N)$, write

$$N_{\pi_1(N)}(H)=\{g\in\pi_1(N):gHg^{-1}=H\}$$

for its normalizer. Bounded normalizer conjecture. The group $\pi_1(N)$ has a non-cyclic, finitely generated subgroup $H$ such that $[N_{\pi_1(N)}(H):H]=\infty$. The source gives no resolution; this is the group-theoretic companion to the bounded virtual bundle conjecture.