Tan–Lei similarity conjecture for cubic superattracting parameter curves

Let $F=F_{a,v}$ be a cubic polynomial in the superattracting period-$p$ parameter curve $4\mathcal{S}_p4$, written in the normal form

$$F_{a,v}(z)=z^3-3a^2z+(2a^3+v).$$

Assume that $F$ is a Misiurewicz map, meaning that its free critical point $-a_F$ is repelling and strictly preperiodic; let $K_F$ be its filled Julia set and let $2a_F$ be the free co-critical point. Tan–Lei similarity conjecture. Under iterated magnification, the connectedness locus around $F$ looks more and more like $K_F$ around $2a_F$, up to a fixed scale change and rotation. This conjecture predicts that the parameter-space geometry of the cubic curve $4\mathcal{S}_p4$ near Misiurewicz maps asymptotically reproduces the corresponding local geometry of the dynamical filled Julia set. The supplied text does not state whether the conjecture has been resolved.