Asymptotic minimal dilatation for fibrations on the magic manifold filling N(1)

Let $N(1)$ be the Dehn filling of the magic 3-manifold along slope $1$. Let $\delta_{1,n}$ denote the least dilatation among the pseudo-Anosov monodromies of fibrations on $N(1)$ in the genus-indexed family specified in the source, and let $\delta(D_4)$ denote the dilatation associated with the braid or mapping class $D_4$.

N(1)-fibration asymptotic conjecture.

$$\lim_{n\to\infty}n\log\delta_{1,n}=2\log\delta(D_4).$$

For large $n$, $\delta_{1,n}$ is achieved by the monodromy of a fibration on $N(1)$.

This predicts both the asymptotic growth rate and the eventual realization of the relevant minimal dilatations by fibrations on the $N(1)$ filling. The supplied context does not define $\delta_{1,n}$ or $D_4$ further and gives no evidence resolving the conjecture.