Edge Monotonicity Conjecture for tessellations of cubic parameter spaces

Let ${\mathfrak p}$ be a parabolic endpoint of edges in ${\bf Tes}_q({\overline{\mathcal S}}_p)$, and let $H_{\mathfrak p}$ be its Type D hyperbolic component. Edge Monotonicity Conjecture. There can be at most four edges with a given parabolic endpoint ${\mathfrak p}$. If there are two or more such edges, then the two edges closest to $H_{\mathfrak p}$ are primary edges, so that the orbit portrait increases monotonically as one crosses into the face containing $H_{\mathfrak p}$. Any further edges are secondary. Furthermore, a primary edge and a secondary edge with the same parabolic endpoint always lie in different escape regions. This conjecture formalizes the observed relation between edge type, orbit-portrait monotonicity, and escape regions; no resolution is given in the source.