Direct-downward arcs conjecture for triangulation graphs

For a given 3-manifold $\mathcal{M}$, let $\phi_n(\mathcal{M})$ denote the fraction of nodes at level $n$ of $\mathscr{P}_1(\mathcal{M})$ that have an arc leading directly down to level $n-1$. Direct-downward arcs conjecture. If $\mathcal{M}$ is the 3-sphere or a closed prime orientable 3-manifold, then

$$\lim_{n \to \infty} \phi_n(\mathcal{M}) = 1.$$

The claim formalizes the observed prevalence of immediately simplifying moves in the census. It is supported by the tabulated data, but the limiting statement remains open.