Tan Lei similarity for cubic polynomial parameter spaces

Let ${\mathcal S}_p$ be the parameter space under consideration, let $F_0\in{\mathcal S}_p$ be a Misiurewicz map, and let $z(F)$ be the unique pre-periodic point near $F_0$ with $z(F_0)=2a_{F_0}$. Define

$$s(F)=2a_F-z(F).$$

Tan Lei Similarity. Under iterated magnification, the connectedness locus around a Misiurewicz map $F\in{\mathcal S}_p$ looks more and more like the filled Julia set of $F$ around the free co-critical point $2a_F$, up to a fixed scale change and rotation. Moreover, the correspondence $F\mapsto s(F)$ is univalent throughout some neighborhood of $F_0$, so that it gives a dynamically defined holomorphic local coordinate; in this coordinate, ${\mathcal S}_p$ near $F_0$ looks more and more like the dynamic plane for $F_0$ near $2a_{F_0}$, without a preliminary scale change or rotation. The conjecture extends Tan Lei's quadratic similarity phenomenon to these parameter spaces; the source presents numerical examples and related ray-order observations, but no resolution.