Normal generation conjecture for pseudo-Anosov monodromies in a fibered cone

Let $F$ be a fibered face of a closed hyperbolic fibered $3$-manifold $M$. A primitive class $(S_\alpha,\psi_\alpha)\in\mathscr{C}_F$ consists of a fiber $S_\alpha$ and its pseudo-Anosov monodromy $\psi_\alpha$.

Normal generation conjecture. For all but finitely many primitive classes $(S_\alpha,\psi_\alpha)\in\mathscr{C}_F$, $\psi_\alpha$ normally generates $\operatorname{Mod}(S_\alpha)$.

This conjecture concerns normal generators among monodromies arising from a fibered face. It is motivated by work of Lanier and Margalit on pseudo-Anosov mapping classes with small stretch factor, and by the question of whether elements in proper normal subgroups must have asymptotic translation length bounded below. The conjecture is presented as open; the paper gives partial evidence from lower bounds for the curve-complex translation lengths of the Torelli groups.