Normalizer Conjecture for closed irreducible 3-manifold groups

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. For a subgroup $H\leq\pi_1(M)$, write

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

for its normalizer. Normalizer Conjecture. The group $\pi_1(M)$ has a non-trivial, finitely generated subgroup $H$ such that $[N_{\pi_1(M)}(H):H]=\infty$. The source gives no resolution; this conjecture forms part of an equivalent set of group-theoretic assertions discussed there.