Monotonicity Conjecture for orbit portraits of cubic polynomials

Let $R$ be a primary parameter ray of co-period $q$ landing at a parabolic point $\mathfrak{p}$, and let $H_\mathfrak{p}$ be the hyperbolic component having $\mathfrak{p}$ as a boundary point. Consider crossing $R$ into the face containing $H_\mathfrak{p}$.

Monotonicity Conjecture. As we cross any primary ray of co-period $q$ into the face which contains $H_\mathfrak{p}$, the period $q$ orbit portrait is replaced by a strictly larger orbit portrait.

This conjecture describes the monotonic behavior of orbit portraits across primary rays in the parameter space of cubic polynomials. The supplied source does not state whether it has been proved or disproved.