Across the Boundary Conjecture for Mandelbrot copies

Let a Mandelbrot copy in ${\mathcal S}_p$ cut across the boundary between two escape regions at a parabolic point ${\mathfrak p}$ of period $q$. The associated dynamic configuration includes periodic rays $\theta_q$, co-periodic rays $\theta$, and their twin rays $\widehat{\theta}$; the marked critical point is $a$ and the free critical point is $-a$. Across the Boundary Conjecture. There are four parameter rays landing on ${\mathfrak p}$, and four corresponding triads in the Julia set for ${\mathfrak p}$. If $p\ne q$, the periodic dynamic rays $\theta_q$ land at the point of period at least two between $a$ and $-a$; the co-periodic rays $\theta$ land at the root point of the hyperbolic component containing $2a$; and the twin rays $\widehat{\theta}$ land on a preimage of $-a$. The midpoint between the periodic landing point and the landing point of the twin rays is always the free critical point $-a$. If ${\mathfrak p}$ is self-dual, as for a Type A component, the twin rays land precisely at the root point of $-2a$. This conjecture seeks a general description of the ray configuration when a Mandelbrot copy crosses an escape-region boundary; the source gives illustrative examples but no resolution.