Uniform critical-orbit comparison for bi-critical irrationally indifferent attractors

Let $f$ be a rational map with a bi-critical irrationally indifferent attractor at $0$. For each active critical point $c$, let $\beta$ denote the angle from $c$ to the other critical point, and let $\mathds{T}_{\alpha,\beta}$ be the associated model map.

Uniform critical-orbit comparison conjecture. There exists a uniform constant $C_{}>0$ such that, for all $k\geq 0$,

$$C_{}^{-1} \left|\mathds{T}_{\alpha,\beta}^{\circ k}(+1)\right| \leq \left|f^{\circ k}(c)\right| \leq C_{}\left|\mathds{T}_{\alpha,\beta}^{\circ k}(+1)\right|.$$

This conjecture predicts that the orbit of each active critical point has, up to a uniform multiplicative constant, the same scale as the corresponding arithmetic-geometric model orbit. The surrounding discussion relates this expected comparison to the geometric similarities between bi-critical and uni-critical near-parabolic renormalisation models; partial progress is known for multi-critical renormalisation, but the stated uniform estimate is not resolved here.