Density of Misiurewicz-model parameters on the common boundary

Let $\mathcal{M}$ and $\mathcal{M}_0$ be the connectivity locus and Thurston set, respectively, for the iterated function system generated by $\mathfrak{s}_-$ and $\mathfrak{s}_+$. Let $g$ be the associated quadratic-like map, and let $\mathsf{A}_\lambda$ be its limit set for parameter $\lambda$. A parameter is Misiurewicz-model when the dynamics of $g$ on $\mathsf{A}_\lambda$ is quasisymmetrically conjugate to the dynamics of $z^2+c$ on its Julia set for some Misiurewicz parameter $c$. Density conjecture. There is a set of parameters $\lambda$, dense in

$$(\partial\mathcal{M}\cap\partial\mathcal{M}_0)\setminus\mathbb{R},$$

for which this quasisymmetric conjugacy holds.

The statement is presented after related results on common-boundary points and gives a proposed description of a dense subset of the common boundary. The supplied text does not establish it or state its resolution.